text:
zold 0.6.2 → 0.6.3

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data/.gitattributes CHANGED
 @@ -5,3 +5,4 @@ 5 5 * text=auto eol=lf 6 6 *.rb ident 7 7 *.xml ident 8 + *.png binary
 @@ -45,8 +45,6 @@ yet, you can run: 45 45 $ssh-keygen -t rsa -b 4096 46 46  47 47 48 - ## Operations 49 - 50 48 ### Remote 51 49 52 50 Each node maintains a list of visible "remote" nodes. @@ -157,42 +155,6 @@ Each HTTP response contains Content-type header. 157 155 158 156 ## Files 159 157 160 - Each wallet is a text file with the name equal to the wallet ID, for example: 161 - 162 - text 163 - 12345678abcdef 164 - AAAAB3NzaC1yc2EAAAADAQABAAABAQCuLuVr4Tl2sXoN5Zb7b6SKMPrVjLxb... 165 - 166 - 34;2017-07-19T21:24:51Z;-560700;98bb82c81735c4ee;for services;SKMPrVjLxbM5oDm0IhniQQy3shF... 167 - 35;2017-07-19T21:25:07Z;-56990;98bb82c81735c4ee;;QCuLuVr4Tl2sXoN5Zb7b6SKMPrVjLxb... 168 - 134;2017-07-19T21:29:11Z;647388;18bb82dd1735b6e9;; 169 - 36;2017-07-19T22:18:43Z;-884733;38ab8fc8e735c4fc;for fun;2sXoN5Zb7b6SKMPrVjLxb7b6SKMPrVjLx... 170 -  171 - 172 - Lines are separated by either CR or CRLF, doesn't matter. There is a 173 - header and a ledger, separated by an empty line. 174 - The header includes two lines: 175 - 176 - * Wallet ID, a 64-bit unsigned integer; 177 - * Public RSA key of the wallet owner. 178 - 179 - The ledger includes transactions, one per line. Each transaction line 180 - contains fields separated by a semi-colon: 181 - 182 - * Transaction ID, an unsigned 16-bit integer; 183 - * Date and time, in [ISO 8601](https://en.wikipedia.org/wiki/ISO_8601); 184 - * Amount (integer); 185 - * Wallet ID of the beneficiary; 186 - * Details: /[a-zA-Z0-9 -.]{1,128}/; 187 - * RSA signature of the sender of "ID;amount;beneficiary;details" text. 188 - 189 - Transactions with positive amount don't 190 - have RSA signatures. Their IDs point to ID fields of corresponding 191 - beneficiaries' wallets. 192 - 193 - The combination "ID+Beneficiary" is unique in the entire wallet. 194 - 195 - The directory .zold is automatically created and contains system data. 196 158 197 159 .zold/remotes is a comma-separated file with a list of remote nodes with 198 160 these columns: data/bin/zold CHANGED  @@ -167,18 +167,8 @@ Available options:" 167 167 require_relative '../lib/zold/commands/push' 168 168 Zold::Push.new(wallets: wallets, remotes: remotes, log: log).run(args) 169 169 when 'score' 170 - require_relative '../lib/zold/score' 171 - raise 'Exactly four args required: host, port, invoice, strength' if args.length != 4 172 - host = args[0] 173 - port = args[1].to_i 174 - invoice = args[2] 175 - strength = args[3].to_i 176 - raise "Invalid strength: #{strength}" if strength <= 0 || strength > 8 177 - score = Zold::Score.new(Time.now, host, port, invoice, strength: strength) 178 - loop do 179 - log.info(score.to_s) 180 - score = score.next 181 - end 170 + require_relative '../lib/zold/commands/calculate' 171 + Zold::Calculate.new(log: log).run(args) 182 172 else 183 173 raise "Command '#{command}' is not supported" 184 174 end  @@ -0,0 +1,8 @@ 1 + #!/bin/bash 2 + set -x 3 + set -e 4 + shopt -s expand_aliases 5 + 6 + alias zold="$1 --ignore-global-config --trace" 7 + 8 + zold score --host=zold.io --port=4096 --invoice=NOSUFFIX@ffffffffffffffff --strength=2 --max=5
 @@ -3,7 +3,7 @@ set -x 3 3 set -e 4 4 shopt -s expand_aliases 5 5 6 - alias zold="$1" 6 + alias zold="$1 --ignore-global-config --trace" 7 7 8 8 zold --help 9 9 declare -a commands=(node create invoice remote pay show fetch clean diff merge propagate pull push)
 @@ -0,0 +1,84 @@ 1 + # Copyright (c) 2018 Yegor Bugayenko 2 + # 3 + # Permission is hereby granted, free of charge, to any person obtaining a copy 4 + # of this software and associated documentation files (the 'Software'), to deal 5 + # in the Software without restriction, including without limitation the rights 6 + # to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 7 + # copies of the Software, and to permit persons to whom the Software is 8 + # furnished to do so, subject to the following conditions: 9 + # 10 + # The above copyright notice and this permission notice shall be included in all 11 + # copies or substantial portions of the Software. 12 + # 13 + # THE SOFTWARE IS PROVIDED 'AS IS', WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 14 + # IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 15 + # FITNESS FOR A PARTICULAR PURPOSE AND NONINFINGEMENT. IN NO EVENT SHALL THE 16 + # AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 17 + # LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 18 + # OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 19 + # SOFTWARE. 20 + 21 + require 'slop' 22 + require_relative '../log' 23 + require_relative '../score' 24 + 25 + # SCORE command. 26 + # Author:: Yegor Bugayenko (yegor256@gmail.com) 27 + # Copyright:: Copyright (c) 2018 Yegor Bugayenko 28 + # License:: MIT 29 + module Zold 30 + # Calculate score 31 + class Calculate 32 + def initialize(log: Log::Quiet.new) 33 + @log = log 34 + end 35 + 36 + def run(args = []) 37 + opts = Slop.parse(args, help: true, suppress_errors: true) do |o| 38 + o.banner = "Usage: zold push [ID...] [options] 39 + Available options:" 40 + o.string '--invoice', 41 + 'The invoice you want to collect money to' 42 + o.integer '--port', 43 + "TCP port to open for the Net (default: #{Remotes::PORT})", 44 + default: Remotes::PORT 45 + o.string '--host', 'Host name (default: 127.0.0.1)', 46 + default: '127.0.0.1' 47 + o.integer '--strength', 48 + "The strength of the score (default: #{Score::STRENGTH})", 49 + default: Score::STRENGTH 50 + o.integer '--max', 51 + 'Maximum value to find and then stop (default: 8)', 52 + default: 8 53 + o.bool '--hide-hash', 'Don\'t print hash', 54 + default: false 55 + o.bool '--help', 'Print instructions' 56 + end 57 + if opts.help? 58 + @log.info(opts.to_s) 59 + return 60 + end 61 + calculate(opts) 62 + end 63 + 64 + def calculate(opts) 65 + start = Time.now 66 + strength = opts[:strength] 67 + raise "Invalid strength: #{strength}" if strength <= 0 || strength > 8 68 + score = Zold::Score.new( 69 + Time.now, opts[:host], opts[:port].to_i, 70 + opts[:invoice], strength: strength 71 + ) 72 + loop do 73 + msg = score.to_s 74 + msg += (score.value > 0 ? ' ' + score.hash : '') unless opts['hide-hash'] 75 + @log.info(msg) 76 + break if score.value >= opts[:max].to_i 77 + score = score.next 78 + end 79 + seconds = (Time.now - start).round(2) 80 + @log.info("Took #{seconds} seconds, #{seconds / score.value} per value") 81 + score 82 + end 83 + end 84 + end
 @@ -66,20 +66,24 @@ Available options:" 66 66 next 67 67 end 68 68 unless response.code == '200' 69 + @remotes.error(r[:host], r[:port]) 69 70 @log.error("#{uri} failed as #{response.code}/#{response.message}") 70 71 next 71 72 end 72 73 json = JSON.parse(response.body)['score'] 73 74 score = Score.parse_json(json) 74 75 unless score.valid? 76 + @remotes.error(r[:host], r[:port]) 75 77 @log.error("#{uri} invalid score: #{score}") 76 78 next 77 79 end 78 80 if score.expired? 81 + @remotes.error(r[:host], r[:port]) 79 82 @log.error("#{uri} expired score: #{score}") 80 83 next 81 84 end 82 85 if score.strength < Score::STRENGTH 86 + @remotes.error(r[:host], r[:port]) 83 87 @log.error("#{uri} score is too weak") 84 88 next 85 89 end
data/lib/zold/score.rb CHANGED
 @@ -96,13 +96,20 @@ module Zold 96 96 ) 97 97 end 98 98 99 + def hash 100 + raise 'Score has zero value, there is no hash' if @suffixes.empty? 101 + @suffixes.reduce(prefix) do |pfx, suffix| 102 + Digest::SHA256.hexdigest(pfx + ' ' + suffix)[0, 63] 103 + end 104 + end 105 + 99 106 def to_text 100 - prefix, bnf = @invoice.split('@') 107 + pfx, bnf = @invoice.split('@') 101 108 [ 102 109 @time.to_i.to_s(16), 103 110 @host, 104 111 @port.to_s(16), 105 - prefix, 112 + pfx, 106 113 bnf, 107 114 @suffixes.join(' ') 108 115 ].join(' ') @@ -127,7 +134,8 @@ module Zold 127 134 invoice: @invoice, 128 135 time: @time.utc.iso8601, 129 136 suffixes: @suffixes, 130 - strength: @strength 137 + strength: @strength, 138 + hash: value.zero? ? nil : hash 131 139 } 132 140 end 133 141 @@ -160,14 +168,12 @@ module Zold 160 168 @time < Time.now - 24 * 60 161 169 end 162 170 171 + def prefix 172 + "#{@time.utc.iso8601} #{@host} #{@port} #{@invoice}" 173 + end 174 + 163 175 def valid? 164 - start = "#{@time.utc.iso8601} #{@host} #{@port} #{@invoice}" 165 - @suffixes.reduce(start) do |prefix, suffix| 166 - hex = Digest::SHA256.hexdigest(prefix + ' ' + suffix) 167 - return false unless hex.end_with?('0' * @strength) 168 - hex[0, 63] 169 - end 170 - true 176 + @suffixes.empty? || hash.end_with?('0' * @strength) 171 177 end 172 178 173 179 def value
 @@ -42,7 +42,7 @@ module Zold 42 42 private 43 43 44 44 def body(id, t) 45 - [id, t.id, t.amount.to_i, t.prefix, t.bnf, t.details].join(' ') 45 + [id, t.id, t.date.utc.iso8601, t.amount.to_i, t.prefix, t.bnf, t.details].join(' ') 46 46 end 47 47 end 48 48 end
data/lib/zold/version.rb CHANGED
 @@ -23,5 +23,5 @@ 23 23 # Copyright:: Copyright (c) 2018 Yegor Bugayenko 24 24 # License:: MIT 25 25 module Zold 26 - VERSION = '0.6.2'.freeze 26 + VERSION = '0.6.3'.freeze 27 27 end
 @@ -0,0 +1,37 @@ 1 + # Copyright (c) 2018 Yegor Bugayenko 2 + # 3 + # Permission is hereby granted, free of charge, to any person obtaining a copy 4 + # of this software and associated documentation files (the 'Software'), to deal 5 + # in the Software without restriction, including without limitation the rights 6 + # to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 7 + # copies of the Software, and to permit persons to whom the Software is 8 + # furnished to do so, subject to the following conditions: 9 + # 10 + # The above copyright notice and this permission notice shall be included in all 11 + # copies or substantial portions of the Software. 12 + # 13 + # THE SOFTWARE IS PROVIDED 'AS IS', WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 14 + # IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 15 + # FITNESS FOR A PARTICULAR PURPOSE AND NONINFINGEMENT. IN NO EVENT SHALL THE 16 + # AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 17 + # LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 18 + # OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 19 + # SOFTWARE. 20 + 21 + require 'minitest/autorun' 22 + require_relative '../../lib/zold/log' 23 + require_relative '../../lib/zold/commands/calculate' 24 + 25 + # SCORE test. 26 + # Author:: Yegor Bugayenko (yegor256@gmail.com) 27 + # Copyright:: Copyright (c) 2018 Yegor Bugayenko 28 + # License:: MIT 29 + class TestCalculate < Minitest::Test 30 + def test_calculates_score 31 + score = Zold::Calculate.new.run( 32 + ['score', '--strength=2', '--max=8', '--invoice=NOSUFFIX@ffffffffffffffff'] 33 + ) 34 + assert(score.valid?) 35 + assert_equal(8, score.value) 36 + end 37 + end
data/test/test_score.rb CHANGED
 @@ -95,4 +95,12 @@ class TestScore < Minitest::Test 95 95 assert(score.valid?) 96 96 assert(!score.expired?) 97 97 end 98 + 99 + def test_correct_number_of_zeroes 100 + score = Zold::Score.new( 101 + Time.now, 'localhost', 443, 102 + 'NOPREFIX@ffffffffffffffff', strength: 3 103 + ).next 104 + assert(score.hash.end_with?('000')) 105 + end 98 106 end
data/wp/.gitignore CHANGED
 @@ -1,3 +1,4 @@ 1 + _minted-wp 1 2 wp.aux 2 3 wp.bcf 3 4 wp.log
 @@ -1,22 +1,28 @@ 1 1 \documentclass[11pt,oneside]{article} 2 2 \usepackage[utf8]{inputenc} 3 3 \usepackage[american]{babel} 4 - % \usepackage[ 5 - % paperwidth=6in, paperheight=9in, 6 - % bindingoffset=0.25in, left=0.75in, right=0.75in, top=0.75in, bottom=1.25in 7 - % ]{geometry} 8 4 \usepackage{setspace} 9 5 \usepackage{microtype} 10 6 \usepackage{mathpazo} % Palantino font 7 + \usepackage{mdframed} 11 8 \usepackage{minted} 12 - \setminted{fontsize=\footnotesize} 9 + \setminted{fontsize=\small} 13 10 \setminted{breaklines} 14 11 \usemintedstyle{bw} 12 + \BeforeBeginEnvironment{minted}{\vspace{6pt}\begin{mdframed}[topline=false,rightline=false,bottomline=false,linecolor=black,linewidth=2pt]} 13 + \AfterEndEnvironment{minted}{\end{mdframed}} 14 + \usepackage{xcolor} 15 + \usepackage{graphicx} 16 + \newcommand\dd[1]{\colorbox{gray!30}{\texttt{#1}}} 15 17 \usepackage{hyperref} 18 + \hypersetup{colorlinks=true,allcolors=blue!40!black} 16 19 \pagestyle{empty} 17 20 \setstretch{1.2} 21 + \setlength{\topskip}{6pt} 22 + \setlength{\parindent}{0pt} % indent first line 23 + \setlength{\parskip}{6pt} % before par 18 24 19 - \title{Zold, Lightweight Crypto Currency} 25 + \title{\includegraphics[scale=0.05]{logo.png}\\Zold: Lightweight Crypto Currency} 20 26 \author{Yegor Bugayenko\\\texttt{yegor@zold.io}} 21 27 22 28 \begin{document} @@ -49,69 +55,101 @@ of existing digital currencies. 49 55 50 56 \section{Principles} 51 57 52 - \textbf{Open source}. 53 - Zold is a command line tool. Its entire code base is open source. 58 + \textbf{Open Source}. 59 + Zold is a command line tool. Its entire code base is open source 60 + and hosted at the GitHub \href{https://github.com/yegor256/zold}{yegor256/zold} 61 + repository. 62 + 63 + \textbf{No Trust}. 64 + The network of communicating nodes maintains wallets of users. 65 + Anyone can add a node to the network. 66 + It is assumed that any node may contain corrupted data, either by mistake or intentionally. 67 + 68 + \textbf{Proof of work}. 69 + Each node, in order to earn trust, must invest its CPU power 70 + and find hash suffixes, performing certain calculations. 71 + 72 + \textbf{No General Ledger}. 73 + There is no central ledger, each wallet has its own personal ledger. 74 + All transactions in each ledger are confirmed by 75 + \href{https://en.wikipedia.org/wiki/RSA_(cryptosystem)}{RSA signatures} 76 + of their owners; 54 77 55 78 \textbf{Capacity}. 56 79 One currency unit is called ZLD. 57 - One ZLD by convention equals to $2^24$ (16,777,216) \emph{zents}. 80 + One ZLD by convention equals to $2^{24}$ \emph{zents} (16,777,216). 58 81 All amounts are stored as signed 64-bit integers. 59 82 Thus, the technical capacity of the currency is 549,755,813,888 ZLD (half a trillion). 60 83 61 - \textbf{Zero wallet}. 62 - There is no mining, the only way to get ZLD is to receive it from someone else. 63 - The wallet with the \texttt{0x00} ID belongs to the 84 + \textbf{Zero Wallet}. 85 + There is no mining,'' the only way to get ZLD is to receive it from someone else. 86 + The wallet with the \dd{0x00} ID belongs to the 64 87 issuer and may have a negative balance. All other wallets 65 - may have only positive balances. 66 - 67 - \textbf{No general ledger}. 68 - There is no central ledger, each wallet has its own personal ledger. 69 - Each transaction in the ledger is confirmed by RSA signature; 70 - 71 - \textbf{No trust}. 72 - The network of communicating nodes maintains wallets of users. 73 - Anyone can add a node to the network. 74 - It is assumed that any node may contain corrupted data, either by mistake or intentionally. 88 + may only have positive balances. 75 89 76 90 \section{Proof of Work} 77 91 78 - Each node calculates its own score. 79 - First, it builds the initial text body, which consists of four parts, 92 + The system consists of nodes (server machines), which maintain the data. 93 + In order to guarantee data consistency among all distributed nodes 94 + there has to be an algorithm of data segregation. 95 + Corrupted data must be detected earlier and filtered out as quickly as possible. 96 + Bitcoin introduced such an algorithm and called it \emph{proof of work}. 97 + 98 + Its fundamental principle is that each block of data must have a special 99 + number attached to it, known as \emph{nonce}, which is rather difficult to calculate, 100 + because it requires a lot of CPU power. It is assumed that at any moment 101 + of time the majority of nodes in the network invest their CPU power into 102 + calculating the nonces for the data that is not corrupted. If and when for any reason 103 + some data gets corrupted, the amount of CPU power a part of the network 104 + decides to invest into its nonces calculation would be smaller than what 105 + the other part of the network invests into legal data. The latter part 106 + will quickly dominate the former and the nodes with corrupted data will 107 + be ostracized and eventually ignored. 108 + 109 + Zold has borrowed this principle, although modified it. We also require 110 + our nodes to invest their CPU power into meaninless and repetative 111 + calculations just to help us identify which part of the network they belong to: 112 + corrupted or not. Each Zold node has to calculate its \emph{trust score}, 113 + which is as big as much CPU power the node has invested into its calculation. 114 + 115 + Similar to Bitcoin nonces we repetatively calculate cryptographic hashes, 116 + looking for consecutive zeros inside them. First, in order to calculate a score, 117 + a node makes the \emph{prefix}, which consists of four parts, 80 118 separated by spaces: 81 119 82 - \begin{itemize} 83 - \item The current timestamp in UTC ISO 8601, 84 - \item The host name or IP address, e.g. \texttf{b2.zold.io}, 85 - \item The TCP port number, 120 + \begin{enumerate} 121 + \item The current timestamp in UTC, in \href{https://en.wikipedia.org/wiki/ISO_8601}{ISO 8601}, 122 + \item The host name or IP address, e.g. \dd{b2.zold.io}, 123 + \item The \href{https://en.wikipedia.org/wiki/Port_(computer_networking)}{TCP port} number, 86 124 \item The invoice. 87 - \end{itemize} 125 + \end{enumerate} 88 126 89 - For example, the body may look like this: 127 + For example, the prefix may look like this: 90 128 91 129 \begin{minted}{text} 92 - 2017-07-19T21:24:51Z b2.zold.io 4096 Yt0lOy6Rgf@0000000000000000 130 + 2018-05-17T03:50:59Z b2.zold.io 4096 THdonv1E@0000000000000000 93 131 \end{minted} 94 132 95 - Then, it attempts to append any 96 - arbitrary text (has to match \texttt{[a-zA-Z0-9]+} regular expression) 97 - to the end of it and to calculate SHA-256 of the text 98 - in the hexadecimal format. For example, this would be the body 99 - with \texttt{abcdef} suffix: 133 + Then, the node attempts to append any arbitrary text, which has to match 134 + \dd{/[a-zA-Z0-9]+/} regular expression, to the end of the prefix and calculates 135 + \href{https://en.wikipedia.org/wiki/SHA-2}{SHA-256 hash} 136 + of the text in the hexadecimal format. For example, this would be the prefix 137 + with the attached \dd{117b1f} suffix: 100 138 101 139 \begin{minted}{text} 102 - 2017-07-19T21:24:51Z b2.zold.io 4096 Yt0lOy6Rgf@0000000000000000 abcdef 140 + 2018-05-17T03:50:59Z b2.zold.io 4096 THdonv1E@0000000000000000 117b1f 103 141 \end{minted} 104 142 105 - The SHA-256 of this body will be: 143 + The hash of this text will be: 106 144 107 145 \begin{minted}{text} 108 - 2017-07-19T21:24:51Z b2.zold.io 4096 Yt0lOy6Rgf@0000000000000000 abcdef 146 + 670baa704726fe2c837c5ca764202adca5ab12c9b90c94d9fb1b8d629000000 109 147 \end{minted} 110 148 111 149 The node attempts to try different sufficies until one of them produces 112 - SHA-256 hash that ends with \texttt{000000} (six zeros). For example, this 113 - suffix \texttt{...} works 114 - (it took about an hour to find it on 2.3GHz Intel Core i7): 150 + a hash that ends with a few tailing zeroes. The one above ends 151 + with six zeroes 152 + (it took three minutes to find it on 2.3GHz Intel Core i7): 115 153 116 154 When the first suffix is found, the score is 1. Then, to 117 155 increase the score by one, the next suffix has to be found, which @@ -119,18 +157,82 @@ can be added to the first 64 characters of the previous hash 119 157 in order to obtain a new hash with trailing zeros, for example: 120 158 121 159 \begin{minted}{text} 122 - 2017-07-19T21:24:51Z b2.zold.io 4096 Yt0lOy6Rgf@0000000000000000 abcdef abcdef 160 + 2018-05-17T03:50:59Z b2.zold.io 4096 THdonv1E@0000000000000000 117b1f 1546e35 161 + \end{minted} 162 + 163 + Produces: 164 + 165 + \begin{minted}{text} 166 + 99dcd18e4bd03004e1205437866b5b68035cc8985240ae52cbd37640a000000 123 167 \end{minted} 124 168 125 169 And so on. 126 170 127 171 The score is valid only when the starting time point is earlier than 128 - current time, but not earlier than 24 hours ago. The strength of the score 172 + the current time, but not earlier than 24 hours ago. The strength of the score 129 173 is the amount of the trailing zeros in the hash. In the example above the 130 - strength was equal to six. 174 + strength is six. 131 175 132 176 \section{Wallets} 133 177 178 + There is no central ledger in Zold. 179 + Each user has \emph{wallets} (any number of them). 180 + Each wallet is an ASCII-text file with the name equal to the wallet ID. 181 + For example, the wallet in the file \dd{12345678abcdef} may include: 182 + 183 + \begin{minted}{text} 184 + 12345678abcdef 185 + AAAAB3NzaC1yc2EAAAADAQABAAABAQCuLuVr4Tl2sXoN5Zb7b6SKMPrVjLxb... 186 + 187 + 34;2017-07-19T21:24:51Z;-560700;Ui0wpLu7;98bb82c81735c4ee;For services;SKMPrVjLxbM5oDm0IhniQQy3shF... 188 + 35;2017-07-19T21:25:07Z;-56990;xksQuJa9;98bb82c81735c4ee;For food;QCuLuVr4Tl2sXoN5Zb7b6SKMPrVjLxb... 189 + 134;2017-07-19T21:29:11Z;647388;kkIZo09s;18bb82dd1735b6e9;-; 190 + 36;2017-07-19T22:18:43Z;-884733;pplIe28s;38ab8fc8e735c4fc;For programming;2sXoN5Zb7b6SKMPrVjLxb7b6SKMPrVjLx... 191 + \end{minted} 192 + 193 + Lines are separated by either CR or CRLF. 194 + There is a header and a ledger, separated by an empty line. 195 + The header includes two lines: 196 + 197 + \begin{enumerate} 198 + \item Wallet ID, a 64-bit unsigned integer in hexadecimal format; 199 + \item Public \href{https://en.wikipedia.org/wiki/RSA_(cryptosystem)}{RSA} 200 + key of the wallet owner, in \href{https://en.wikipedia.org/wiki/Base64}{Base64}. 201 + \end{enumerate} 202 + 203 + The ledger includes transactions, one per line. Each transaction line 204 + contains fields separated by a semi-colon: 205 + 206 + \begin{enumerate} 207 + \item Transaction ID, an unsigned 16-bit integer; 208 + \item Date and time, in \href{https://en.wikipedia.org/wiki/ISO_8601}{ISO 8601} format; 209 + \item Amount, a signed 64-bit integer; 210 + \item Payment prefix, 8-32 symbols; 211 + \item Wallet ID of the beneficiary; 212 + \item Details, matching \dd{/[a-zA-Z0-9 -.]\{1,128\}/}; 213 + \item \href{https://en.wikipedia.org/wiki/RSA_(cryptosystem)}{RSA} signature, 214 + 684 symbols in \href{https://en.wikipedia.org/wiki/Base64}{Base64}. 215 + \end{enumerate} 216 + 217 + Transactions with positive amount don't have signatures. 218 + Their IDs point to ID fields of corresponding beneficiaries' wallets. 219 + 220 + The combination \dd{ID+Beneficiary} is unique in the entire wallet. 221 + 222 + The \href{https://en.wikipedia.org/wiki/RSA_(cryptosystem)}{RSA} 223 + signature is calculated using the private RSA key of the 224 + wallet and the following fields of transaction, separated by spaces: 225 + 226 + \begin{enumerate} 227 + \item Wallet ID; 228 + \item Transaction ID, an unsigned 16-bit integer; 229 + \item Date and time, in ISO 8601 format; 230 + \item Amount, a signed 64-bit integer; 231 + \item Payment prefix; 232 + \item Wallet ID of the beneficiary; 233 + \item Details, matching \dd{/[a-zA-Z0-9 -.]{1,128}/}. 234 + \end{enumerate} 235 + 134 236 \section{Push} 135 237 136 238 \section{Fetch and Merge}


text:
# Writing linear equations

### What is a Linear Function?

Linear functions are algebraic equations whose graphs are straight lines with unique values for their slope and y-intercepts.

### Learning Objectives

Describe the parts and characteristics of a linear function

### Key Takeaways

#### Key Points

• A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
• A function is a relation with the property that each input is related to exactly one output.
• A relation is a set of ordered pairs.
• The graph of a linear function is a straight line, but a vertical line is not the graph of a function.
• All linear functions are written as equations and are characterized by their slope and $y$-intercept.

#### Key Terms

• relation: A collection of ordered pairs.
• variable: A symbol that represents a quantity in a mathematical expression, as used in many sciences.
• linear function: An algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
• function: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

#### What is a Linear Function?

A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  It is linear: the exponent of the $x$ term is a one (first power), and it follows the definition of a function: for each input ($x$) there is exactly one output ($y$).  Also, its graph is a straight line.

### Graphs of Linear Functions

The origin of the name &#;linear&#; comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In the linear function graphs below, the constant, $m$, determines the slope or gradient of that line, and the constant term, $b$, determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.

Graphs of linear functions: The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  The blue line has a positive slope of $\frac{1}{2}$ and a $y$-intercept of $-3$; the red line has a negative slope of $-1$ and a $y$-intercept of $5$.

### Vertical and Horizontal Lines

Vertical lines have an undefined slope, and cannot be represented in the form $y=mx+b$, but instead as an equation of the form $x=c$ for a constant $c$, because the vertical line intersects a value on the $x$-axis, $c$.  For example, the graph of the equation $x=4$ includes the same input value of $4$ for all points on the line, but would have different output values, such as $(4,-2),(4,0),(4,1),(4,5),$ etcetera. Vertical lines are NOT functions, however, since each input is related to more than one output.

Horizontal lines have a slope of zero and is represented by the form, $y=b$, where $b$ is the $y$-intercept.  A graph of the equation $y=6$ includes the same output value of 6 for all input values on the line, such as $(-2,6),(0,6),(2,6),(6,6)$, etcetera.  Horizontal lines ARE functions because the relation (set of points) has the characteristic that each input is related to exactly one output.

### Slope

Slope describes the direction and steepness of a line, and can be calculated given two points on the line.

### Learning Objectives

Calculate the slope of a line using &#;rise over run&#; and identify the role of slope in a linear equation

### Key Takeaways

#### Key Points

• The slope of a line is a number that describes both the direction and the steepness of the line; its sign indicates the direction, while its magnitude indicates the steepness.
• The ratio of the rise to the run is the slope of a line, $m = \frac{rise}{run}$.
• The slope of a line can be calculated with the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are points on the line.

#### Key Terms

• steepness: The rate at which a function is deviating from a reference.
• direction: Increasing, decreasing, horizontal or vertical.

### Slope

In mathematics, the slope of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter $m$. Recall the slop-intercept form of a line, $y = mx + b$. Putting the equation of a line into this form gives you the slope ($m$) of a line, and its $y$-intercept ($b$). We will now discuss the interpretation of $m$, and how to calculate $m$ for a given line.

The direction of a line is either increasing, decreasing, horizontal or vertical. A line is increasing if it goes up from left to right which implies that the slope is positive ($m > 0$). A line is decreasing if it goes down from left to right and the slope is negative ($m < 0$). If a line is horizontal the slope is zero and is a constant function ($y=c$). If a line is vertical the slope is undefined.

Slopes of Lines: The slope of a line can be positive, negative, zero, or undefined.

The steepness, or incline, of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.

### Calculating Slope

Slope is calculated by finding the ratio of the &#;vertical change&#; to the &#;horizontal change&#; between any two distinct points on a line. This ratio is represented by a quotient (&#;rise over run&#;), and gives the same number for any two distinct points on the same line. It is represented by $m = \frac{rise}{run}$.

Visualization of Slope: The slope of a line is calculated as &#;rise over run.&#;

Mathematically, the slope m of the line is:

$\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Two points on the line are required to find $m$. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, take a look at the graph below and note how the &#;rise&#; of slope is given by the difference in the $y$-values of the two points, and the &#;run&#; is given by the difference in the $x$-values.

Slope Represented Graphically: The slope $m =\frac{y_{2} &#; y_{1}}{x_{2} &#; x_{1}}$ is calculated from the two points $\left( x_1,y_1 \right)$ and $\left( x_2,y_2 \right)$.

Now we’ll look at some graphs on a coordinate grid to find their slopes. In many cases, we can find slope by simply counting out the rise and the run. We start by locating two points on the line. If possible, we try to choose points with coordinates that are integers to make our calculations easier.

### Example

Find the slope of the line shown on the coordinate plane below.

Find the slope of the line: Notice the line is increasing so make sure to look for a slope that is positive.

Locate two points on the graph, choosing points whose coordinates are integers. We will use $(0, -3)$ and $(5, 1)$. Starting with the point on the left, $(0, -3)$, sketch a right triangle, going from the first point to the second point, $(5, 1)$.

Identify points on the line: Draw a triangle to help identify the rise and run.

Count the rise on the vertical leg of the triangle: $4$ units.

Count the run on the horizontal leg of the triangle: $5$ units.

Use the slope formula to take the ratio of rise over run:

\displaystyle \begin{align} m &= \frac{rise}{run} \\ &= \frac{4}{5} \end{align}

The slope of the line is $\frac{4}{5}$. Notice that the slope is positive since the line slants upward from left to right.

### Example

Find the slope of the line shown on the coordinate plane below.

Find the slope of the line: We can see the slope is decreasing, so be sure to look for a negative slope.

Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use $(0, 5)$ and $(3, 3)$.

Identify two points on the line: The points $(0, 5)$ and $(3, 3)$ are on the line.

$\displaystyle m =\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Let $(x_1, y_1)$ be the point $(0, 5)$, and $(x_2, y_2)$ be the point $(3, 3)$.

Plugging the corresponding values into the slope formula, we get:

\displaystyle \begin{align} m &= \frac{}{} \\ &= \frac{-2}{3} \end{align}

The slope of the line is $- \frac{2}{3}$. Notice that the slope is negative since the line slants downward from left to right.

### Direct and Inverse Variation

Two variables in direct variation have a linear relationship, while variables in inverse variation do not.

### Learning Objectives

Recognize examples of functions that vary directly and inversely

### Key Takeaways

#### Key Points

• Two variables that change proportionally to one another are said to be in direct variation.
• The relationship between two directly proportionate variable can be represented by a linear equation in slope -intercept form, and is easily modeled using a linear graph.
• Inverse variation is the opposite of direct variation; two variables are said to be inversely proportional when a change is performed on one variable and the opposite happens to the
other.
• The relationship between two inversely proportionate variables cannot be represented by a linear equation, and its graphical representation is not a line, but a hyperbola.

#### Key Terms

• hyperbola: A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.
• proportional: At a constant ratio. Two magnitudes (numbers) are said to be proportional if the second varies in a direct relation arithmetically to the first.

### Direct Variation

Simply put, two variables are in direct variation when the same thing that happens to one variable happens to the other. If $x$ and $y$ are in direct variation, and $x$ is doubled, then $y$ would also be doubled. The two variables may be considered directly proportional.

For example, a toothbrush costs $2$ dollars. Purchasing $5$ toothbrushes would cost $10$ dollars, and purchasing $10$ toothbrushes would $20$ cost dollars. Thus we can say that the cost varies directly as the value of toothbrushes.

Direct variation is represented by a linear equation, and can be modeled by graphing a line. Since we know that the relationship between two values is constant, we can give their relationship with:

$\displaystyle \frac{y}{x} = k$

Where $k$ is a constant.

Rewriting this equation by multiplying both sides by $x$ yields:

$\displaystyle y = kx$

Notice that this is a linear equation in slope-intercept form, where the $y$-intercept $b$ is equal to $0$.

Thus, any line passing through the origin represents a direct variation between $x$ and $y$:

Directly Proportional Variables: The graph of $y = kx$ demonstrates an example of direct variation between two variables.

Revisiting the example with toothbrushes and dollars, we can define the $x$-axis as number of toothbrushes and the $y$-axis as number of dollars. Doing so, the variables would abide by the relationship:

$\displaystyle \frac{y}{x} = 2$

Any augmentation of one variable would lead to an equal augmentation of the other. For example, doubling $y$ would result in the doubling of $x$.

### Inverse Variation

Inverse variation is the opposite of direct variation. In the case of inverse variation, the increase of one variable leads to the decrease of another. In fact, two variables are said to be inversely proportional when an operation of change is performed on one variable and the opposite happens to the other. For example, if $x$ and $y$ are inversely proportional, if $x$ is doubled, then $y$ is halved.

As an example, the time taken for a journey is inversely proportional to the speed of travel. If your car travels at a greater speed, the journey to your destination will be shorter.

Knowing that the relationship between the two variables is constant, we can show that their relationship is:

$\displaystyle yx = k$

Where $k$ is a constant known as the constant of proportionality. Note that as long as $k$ is not equal to $0$, neither $x$ nor $y$ can ever equal $0$ either. We can rearrange the above equation to place the variables on opposite sides:

$\displaystyle y=\frac{k}{x}$

Notice that this is not a linear equation. It is impossible to put it in slope-intercept form. Thus, an inverse relationship cannot be represented by a line with constant slope. Inverse variation can be illustrated with a graph in the shape of a hyperbola, pictured below.

Inversely Proportional Function: An inversely proportional relationship between two variables is represented graphically by a hyperbola.

### Zeroes of Linear Functions

A zero, or $x$-intercept, is the point at which a linear function&#;s value will equal zero.

### Learning Objectives

Practice finding the zeros of linear functions

### Key Takeaways

#### Key Points

• A zero is a point at which a function &#;s value will be equal to zero. Its coordinates are $(x,0)$, where $x$ is equal to the zero of the graph.
• Zeros can be observed graphically or solved for algebraically.
• A linear function can have none, one, or infinitely many zeros. If the function is a horizontal line ( slope = $0$), it will have no zeros unless its equation is $y=0$, in which case it will have infinitely many. If the line is non-horizontal, it will have one zero.

#### Key Terms

• zero: Also known as a root; an $x$ value at which the function of $x$ is equal to $0$.
• linear function: An algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
• y-intercept: A point at which a line crosses the $y$-axis of a Cartesian grid.

The graph of a linear function is a straight line. Graphically, where the line crosses the $x$-axis, is called a zero, or root.  Algebraically, a zero is an $x$ value at which the function of $x$ is equal to $0$.  Linear functions can have none, one, or infinitely many zeros.  If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the $x$-axis.  If the horizontal line overlaps the $x$-axis, (goes through the $y$-axis at zero) then there are infinitely many zeros, since the line intersects the $x$-axis multiple times.  Finally, if the line is vertical or has a slope, then there will be only one zero.

### Finding the Zeros of Linear Functions Graphically

Zeros can be observed graphically.  An $x$-intercept, or zero, is a property of many functions. Because the $x$-intercept (zero) is a point at which the function crosses the $x$-axis, it will have the value $(x,0)$, where $x$ is the zero.

All lines, with a value for the slope, will have one zero.  To find the zero of a linear function, simply find the point where the line crosses the $x$-axis.

Zeros of linear functions: The blue line, $y=\frac{1}{2}x+2$, has a zero at $(-4,0)$; the red line, $y=-x+5$, has a zero at $(5,0)$.  Since each line has a value for the slope, each line has exactly one zero.

### Finding the Zeros of Linear Functions Algebraically

To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.

The zero from solving the linear function above graphically must match solving the same function algebraically.

### Example: Find the zero of $y=\frac{1}{2}x+2$ algebraically

First, substitute $0$ for $y$:

$\displaystyle 0=\frac{1}{2}x+2$

Next, solve for $x$. Subtract $2$ and then multiply by $2$, to obtain:

\displaystyle \begin{align} \frac{1}{2}x&=-2\\ x&=-4 \end{align}

The zero is $(-4,0)$.  This is the same zero that was found using the graphing method.

### Slope-Intercept Equations

The slope-intercept form of a line summarizes the information necessary to quickly construct its graph.

### Learning Objectives

Convert linear equations to slope-intercept form and explain why it is useful

### Key Takeaways

#### Key Points

• The slope -intercept form of a line is given by $y = mx + b$ where $m$ is the slope of the line and $b$ is the $y$-intercept.
• The constant $b$ is known as the $y$-intercept.  From slope- intercept form, when $x=0$, $y=b$, and the point $(0,b)$ is the unique point on the line also on the $y$-axis.
• To graph a line in slope-intercept form, first plot the $y$-intercept, then use the value of the slope to locate a second point on the line.  If the value of the slope is an integer, use a $1$ for the denominator.
• Use algebra to solve for $y$ if the equation is not written in slope-intercept form. Only then can the value of the slope and $y$-intercept be located from the equation accurately.

#### Key Terms

• slope: The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.
• y-intercept: A point at which a line crosses the $y$-axis of a Cartesian grid.

### Slope-Intercept Form

One of the most common representations for a line is with the slope-intercept form. Such an equation is given by $y=mx+b$, where $x$ and $y$ are variables and $m$ and $b$ are constants.  When written in this form, the constant $m$ is the value of the slope and $b$ is the $y$-intercept.  Note that if $m$ is $0$, then $y=b$ represents a horizontal line. Note that this equation does not allow for vertical lines, since that would require that $m$ be infinite (undefined).  However, a vertical line is defined by the equation $x=c$ for some constant $c$.

### Converting an Equation to Slope-Intercept Form

Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.  This assists in finding solutions to various problems, such as graphing, comparing two lines to determine if they are parallel or perpendicular and solving a system of equations.

### Example

Let&#;s write an equation in slope-intercept form with $m=-\frac{2}{3}$, and $b=3$.  Simply substitute the values into the slope-intercept form to obtain:

$\displaystyle y=-\frac{2}{3}x+3$

If an equation is not in slope-intercept form, solve for $y$ and rewrite the equation.

### Example

Let&#;s write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept. To solve the equation for $y$, first subtract $3x$ from both sides of the equation to get:

$\displaystyle 2y=-3x-4$

Then divide both sides of the equation by $2$ to obtain:

$\displaystyle y=\frac{1}{2}(-3x-4)$

Which simplifies to $y=-\frac{3}{2}x-2$.  Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.

### Graphing an Equation in Slope-Intercept Form

We begin by constructing the graph of the equation in the previous example.

### Example

We construct the graph the line $y=-\frac{3}{2}x-2$ using the slope-intercept method. We begin by plotting the $y$-intercept $b=-2$, whose coordinates are $(0,-2)$.  The value of the slope dictates where to place the next point.

Since the value of the slope is $\frac{-3}{2}$, the rise is $-3$ and the run is $2$.  This means that from the $y$-intercept, $(0,-2)$, move $3$ units down, and move $2$ units right.  Thus we arrive at the point $(2,-5)$ on the line. If the negative sign is placed with the denominator instead the slope would be written as $\frac{3}{-2}$, we can instead move up $3$ units and left $2$ units from the $y$-intercept to arrive at the point $(-2,1)$, also on the line.

Slope-intercept graph: Graph of the line $y=-\frac{3}{2}x-2$.

### Example

Let&#;s graph the equation $12x-6y-6=0$. First we solve the equation for $y$ by subtracting $12x$ to obtain:

$\displaystyle -6y-6=x$

Next, add $6$ to get:

$\displaystyle -6y=x+6$

Finally, divide all terms by $-6$ to get the slope-intercept form:

$\displaystyle y=2x-1$

The slope is $2$, and the $y$-intercept is $-1$.  Using this information, graphing is easy.  Start by plotting the $y$-intercept $(0,-1)$, then use the value of the slope, $\frac{2}{1}$, to move up $2$ units and right $1$ unit.

Slope-intercept graph: Graph of the line $y=2x-1$.

### Point-Slope Equations

The point-slope equation is another way to represent a line; only the slope and a single point are needed.

### Learning Objectives

Use point-slope form to find the equation of a line passing through two points and verify that it is equivalent to the slope-intercept form of the equation

### Key Takeaways

#### Key Points

• The point &#; slope equation is given by $y-y_{1}=m(x-x_{1})$, where $(x_{1}, y_{1})$ is any point on the line, and $m$ is the slope of the line.
• The point-slope equation requires that there is at least one point and the slope. If there are two points and no slope, the slope can be calculated from the two points first and then choose one of the two points to write the equation.
• The point-slope equation and slope-intercept equations are equivalent.  It can be shown that given a point $(x_{1}, y_{1})$ and slope $m$, the $y$-intercept ($b$) in the slope-intercept equation is $y_{1}-mx_{1}$.

#### Key Terms

• point-slope equation: An equation of a line given a point $(x_{1}, y_{1})$ and a slope $m$: $y-y_{1}=m(x-x_{1})$.

### Point-Slope Equation

The point-slope equation is a way of describing the equation of a line. The point-slope form is ideal if you are given the slope and only one point, or if you are given two points and do not know what the $y$-intercept is. Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:

$\displaystyle y-y_{1}=m(x-x_{1})$

### Verify Point-Slope Form is Equivalent to Slope-Intercept Form

To show that these two equations are equivalent, choose a generic point $(x_{1}, y_{1})$. Plug in the generic point into the equation $y=mx+b$.  The equation is now, $y_{1}=mx_{1}+b$, giving us the ordered pair,$(x_{1}, mx_{1}+b)$. Then plug this point into the point-slope equation and solve for $y$ to get:

$\displaystyle y-(mx_{1}+b)=m(x-x_{1})$

Distribute the negative sign through and distribute $m$ through $(x-x_{1})$:

$\displaystyle y-mx_{1}-b=mx-mx_{1}$

Add $mx_{1}$ to both sides:

$\displaystyle y-mx_{1}+mx_{1}-b=mx-mx_{1}+mx_{1}$

Combine like terms:

$\displaystyle y-b=mx$

Add $b$ to both sides:

$\displaystyle y-b+b=mx+b$

Combine like terms:

$\displaystyle y=mx+b$

Therefore, the two equations are equivalent and either one can express an equation of a line depending on what information is given in the problem or what type of equation is requested in the problem.

### Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form

Write the equation of the line in point-slope form:

$\displaystyle y-1=-4(x-2)$

To switch this equation into slope-intercept form, solve the equation for $y$:

$\displaystyle y-1=-4(x-2)$

Distribute $-4$:

$\displaystyle y-1=-4x+8$

Add $1$ to both sides:

$\displaystyle y=-4x+9$

The equation has the same meaning whichever form it is in, and produces the same graph.

Line graph: Graph of the line $y-1=-4(x-2)$, through the point $(2,1)$ with slope of $-4$, as well as the slope-intercept form, $y=-4x+9$.

### Example: Write the equation of a line in point-slope form, given point $(-3,6)$ and point $(1,2)$, and convert to slope-intercept form

Since we have two points, but no slope, we must first find the slope:

$\displaystyle m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Substituting the values of the points:

\displaystyle \begin{align} m&=\frac{}{1-(-3)}\\&=\frac{-8}{4}\\&=-2 \end{align}

Now choose either of the two points, such as $(-3,6)$. Plug this point and the calculated slope into the point-slope equation to get:

$\displaystyle y-6=-2[x-(-3)]$

Be careful if one of the coordinates is a negative.  Distributing the negative sign through the parentheses, the final equation is:

$\displaystyle y-6=-2(x+3)$

If you chose the other point, the equation would be: $y+2=-2(x-1)$ and either answer is correct.

Next distribute $-2$:

$\displaystyle y-6=-2x-6$

Add $6$ to both sides:

$\displaystyle y=-2x$

Again, the two forms of the equations are equivalent to each other and produce the same line.  The only difference is the form that they are written in.

### Linear Equations in Standard Form

A linear equation written in standard form makes it easy to calculate the zero, or $x$-intercept, of the equation.

### Learning Objectives

Explain the process and usefulness of converting linear equations to standard form

### Key Takeaways

#### Key Points

• The standard form of a linear equation is written as: $Ax + By = C$.
• The standard form is useful in calculating the zero of an equation. For a linear equation in standard form, if $A$ is nonzero, then the $x$-intercept occurs at $x = \frac{C}{A}$.

#### Key Terms

• zero: Also known as a root, a zero is an $x$-value at which the function of $x$ is equal to 0.
• slope-intercept form: A linear equation written in the form $y = mx + b$.
• y-intercept: A point at which a line crosses the y-axis of a Cartesian grid.

### Standard Form

Standard form is another way of arranging a linear equation. In the standard form, a linear equation is written as:

$\displaystyle Ax + By = C$

where $A$ and $B$ are both not equal to zero. The equation is usually written so that $A \geq 0$, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the standard form.

For example, consider an equation in slope -intercept form: $y = x +5$. In order to write this in standard form, note that we must move the term containing $x$ to the left side of the equation. We add $12x$ to both sides:

$\displaystyle y + 12x = 5$

The equation is now in standard form.

### Using Standard Form to Find Zeroes

Recall that a zero is a point at which a function &#;s value will be equal to zero ($y=0$), and is the $x$-intercept of the function. We know that the y-intercept of a linear equation can easily be found by putting the equation in slope-intercept form. However, the zero of the equation is not immediately obvious when the linear equation is in this form. However, the zero, or $x$-intercept of a linear equation can easily be found by putting it into standard form.

For a linear equation in standard form, if $A$ is nonzero, then the $x$-intercept occurs at $x = \frac{C}{A}$.

For example, consider the equation $y + 12x = 5$.

In this equation, the value of $A$ is 1, and the value of $C$ is 5. Therefore, the zero of the equation occurs at $x = \frac{5}{1} = 5$. The zero is the point $(5, 0)$.

Note that the $y$-intercept and slope can also be calculated using the coefficients and constant of the standard form equation. If $B$ is non-zero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where $x$ is zero), is $\frac{C}{B}$, and the slope of the line is $-\frac{A}{B}$.

### Example: Find the zero of the equation $3(y - 2) = \frac{1}{4}x +3$

We must write the equation in standard form, $Ax + By = C$, which means getting the $x$ and $y$ terms on the left side, and the constants on the right side of the equation.

Distribute the 3 on the left side:

$\displaystyle 3y - 6 = \frac{1}{4}x +3$

$\displaystyle 3y = \frac{1}{4}x + 9$

Subtract $\frac{1}{4}x$ from both sides:

$\displaystyle 3y - \frac{1}{4}x = 9$

Rearrange to $Ax + By = C$:

$\displaystyle - \frac{1}{4}x+3y = 9$

The equation is in standard form, and we can substitute the values for $A$ and $C$ into the formula for the zero:

\displaystyle \begin{align} x &= \frac{C}{A} \\&= \frac{9}{-\frac{1}{4}} \\&= \end{align}

The zero is $(, 0)$.

Sours: https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-linear-functions/

## Writing Systems of Linear Equations from Word Problems

Some word problems require the use of systems of linear equations . Here are clues to know when a word problem requires you to write a system of linear equations:

(i) There are two different quantities involved: for instance, the number of adults and the number of children, the number of large boxes and the number of small boxes, etc.

(ii) There is a value associated with each quantity: for instance, the price of an adult ticket or a children's ticket, or the number of items in a large box as opposed to a small box.

Such problems often require you to write two different linear equations in two variables. Typically, one equation will relate the number of quantities (people or boxes) and the other equation will relate the values (price of tickets or number of items in the boxes).

Here are some steps to follow:

1. Understand the problem.

Understand all the words used in stating the problem.

Understand what you are asked to find.

Familiarize the problem situation.

2. Translate the problem to an equation.

Assign a variable (or variables) to represent the unknown.

Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Use substitution , elimination or graphing method to solve the problem.

Example:

The cost of admission to a popular music concert was for children and adults. The admission was for children and adults in another music concert. How much was the admission for each child and adult?

. Understand the problem:

. Translate the problem to an equation.

Let represent the admission cost for each child.

That is, .

The admission cost for 8 children plus 3 adults is equal to \$

That is, .

. Carry out the plan and solve the problem.

Subtract the second equation from the first.

Substitute for in .

Therefore, the cost of admission for each child is and each adult is .

Sours: https://www.varsitytutors.com/hotmath/hotmath_help/topics/writing-systems-of-linear-equations-from-word-problems

## How to Write Linear Equations

In this article, you learn how to write the equation of the lines by using their slope and one point or using two points on the line.

### Step by step guide to writing linear equations

• The equation of a line in slope intercept form is: $$\color{blue}{y=mx+b}$$
• Identify the slope.
• Find the $$y$$–intercept. This can be done by substituting the slope and the coordinates of a point $$(x, y)$$ on the line.

### Writing Linear Equations &#; Example 1:

What is the equation of the line that passes through $$(1, -2)$$ and has a slope of $$6$$?

Solution:

The general slope-intercept form of the equation of a line is $$y=mx+b$$, where $$m$$ is the slope and $$b$$ is the $$y$$-intercept.
By substitution of the given point and given slope, we have: $$-2=(1)(6)+b$$
So, $$b= =-8$$, and the required equation is $$y=6x-8$$.

### Writing Linear Equations &#; Example 2:

Write the equation of the line through $$(1, 1)$$ and $$(-1, 3)$$.

Solution:

Slop $$= \frac{y_{2}- y_{1}}{x_{2} &#; x_{1} }=\frac{3- 1}{ 1}=\frac{2}{-2}=-1 → m=-1$$
To find the value of $$b$$, you can use either points. The answer will be the same: $$y=-x+b$$
$$(1,1) →1=-1+b→b=2$$
$$(-1,3)→3=-(-1)+b→b=2$$
The equation of the line is: $$y=-x+2$$

### Writing Linear Equations &#; Example 3:

What is the equation of the line that passes through $$(2,–2)$$ and has a slope of $$7$$?

Solution:

The general slope-intercept form of the equation of a line is $$y=mx+b$$, where $$m$$ is the slope and $$b$$ is the $$y-$$intercept.
By substitution of the given point and given slope, we have: $$-2=(7)(2)+b$$
So, $$b= –=\$$, and the required equation is $$y=7x$$.

### Writing Linear Equations &#; Example 4:

Write the equation of the line through $$(2,1)$$ and $$(-1,4)$$.

Solution:

Slop $$= \frac{y_{2}- y_{1}}{x_{2} &#; x_{1} }=\frac{4- 1}{ 2}=\frac{3}{-3}=\ -1 → m= \ -1$$
To find the value of $$b$$, you can use either points. The answer will be the same: $$y= \ -x+b$$
$$(2,1) →1=-2+b→b=3$$
$$(-1,4)→4=-(-1)+b→b=3$$
The equation of the line is: $$y= \ -x+3$$

### Write the slope–intercept form of the equation of the line through the given points.

1. $$\color{blue}{through: (– 4, – 2), (– 3, 5)}$$
2. $$\color{blue}{through: (5, 4), (– 4, 3) }$$
3. $$\color{blue}{through: (0, – 2), (– 5, 3) }$$
4. $$\color{blue}{through: (– 1, 1), (– 2, 6) }$$
5. $$\color{blue}{through: (0, 3), (– 4, – 1) }$$
6. $$\color{blue}{through: (0, 2), (1, – 3) }$$

1. $$\color{blue}{y = 7x + 26}$$
2. $$\color{blue}{y = \frac{1}{9} x + \frac{31}{9}}$$
3. $$\color{blue}{y = – x – 2}$$
4. $$\color{blue}{y = &#; 5x – 4}$$
5. $$\color{blue}{y = x + 3}$$
6. $$\color{blue}{y = – 5x + 2}$$

### "How to Write Linear Equations" Tags

How to Write Linear EquationsWriting Linear Equations

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## Linear Equations

A linear equation is an equation for a straight line

### These are all linear equations:

 y = 2x + 1 5x = 6 + 3y y/2 = 3 − x

Let us look more closely at one example:

### Example: y = 2x + 1 is a linear equation:

The graph of y = 2x+1 is a straight line

• When x increases, y increases twice as fast, so we need 2x
• When x is 0, y is already 1. So +1 is also needed
• And so: y = 2x + 1

Here are some example values:

xy = 2x + 1
-1y = 2 × (-1) + 1 = -1
0y = 2 × 0 + 1 = 1
1y = 2 × 1 + 1 = 3
2y = 2 × 2 + 1 = 5

Check for yourself that those points are part of the line above!

### Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

### Examples: These are linear equations:

But the variables (like "x" or "y") in Linear Equations do NOT have:

### Examples: These are NOT linear equations:

 y2 − 2 = 0 3√x − y = 6 x3/2 = 16

### Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

### Example: y = 2x + 1

• Slope: m = 2
• Intercept: b = 1

### Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

 y − y1 = m(x − x1)

### Example: y − 3 = (¼)(x − 2)

It is in the form y − y1 = m(x − x1) where:

• y1 = 3
• m = ¼
• x1 = 2

### General Form

And there is also the General Form of the equation of a straight line:

 Ax + By + C = 0 (A and B cannot both be 0)

### Example: 3x + 2y − 4 = 0

It is in the form Ax + By + C = 0 where:

There are other, less common forms as well.

### As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

 y = 2x − 3 f(x) = 2x − 3 These are the same!

And functions are not always written using f(x):

 y = 2x − 3 w(u) = 2u − 3 h(z) = 2z − 3 These are also the same!

### The Identity Function

There is a special linear function called the "Identity Function":

f(x) = x

And here is its graph:

It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is identical to what goes in:

InOut
00
55
−2−2
etcetc

### Constant Functions

Another special type of linear function is the Constant Function it is a horizontal line:

f(x) = C

No matter what value of "x", f(x) is always equal to some constant value.

### Using Linear Equations

You may like to read some of the things you can do with lines:

Gradient (Slope) of a Straight LineY Intercept of a Straight LineDistance Between 2 PointsFinding Intercepts From an EquationGraph MenuAlgebra Menu

Sours: https://www.mathsisfun.com/algebra/linear-equations.html
Writing A Linear Equation From A Function Table

## How to Write Linear Equations in Algebra

Note that the standard form of a linear equation is:

Where m = slope and b = y-intercept.

Calculate the slope of the line. The slope can be found by selecting two points on the line, determining the vertical rise and the horizontal run between the points and dividing them. For example, if (3,4) and (5,6) are on the line, the slope between them would be (5 - 3) / (6 - 4), simplified to (2) / (2), simplified to 1. Include negative values, since slopes can be positive or negative.

Determine or calculate the y-intercept of the line. The y-intercept is the y-coordinate of the point where the line passes through the y-axis of the coordinate plane. For example, if the point of intersection with the y-axis is (0,5), the y-intercept would be 5. The y-intercept can be found by physically locating it on the graph or by locating the given point on the line that has an x-coordinate of 0. That point is the point of intersection. The y-intercept will be positive if it intersects the y-axis above the x-axis or negative if it intersects below the x-axis.

Write the equation y = mx + b, substituting the values for m and b you calculated or determined. The m will be your slope, and the b will be your y-intercept. Leave the y and x variables in the equation as letter variables. Include the sign of the numbers you plug in. For example, if I discovered my slope to be -3 and my y-intercept to be 5, my linear equation would be y = -3x + 5. The linear equation is complete and correctly written when the (m) and (b) are properly incorporated into the equation.

#### Tips

• Factor negative signs into the linear equation carefully. If b = -8 and m = 5, the algebraic linear equation would be written y = 5x + (- 8), or simplified, y = 5x - 8.

When in doubt, check your work.

Sours: https://sciencing.com/write-linear-equations-algebrahtml

## Writing linear equations using the slope-intercept form

An equation in the slope-intercept form is written as

$$y=mx+b$$

Where m is the slope of the line and b is the y-intercept. You can use this equation to write an equation if you know the slope and the y-intercept.

Example

Find the equation of the line

Choose two points that are on the line

Calculate the slope between the two points

$$m=\frac{y_{2}\, -y_{1}}{x_{2}\, -x_{1}}=\frac{\left (-1 \right )-3}{3-\left ( -3 \right )}=\frac{-4}{6}=\frac{-2}{3}$$

We can find the b-value, the y-intercept, by looking at the graph

b = 1

We've got a value for m and a value for b. This gives us the linear function

$$y=-\frac{2}{3}x+1$$

In many cases the value of b is not as easily read. In those cases, or if you're uncertain whether the line actually crosses the y-axis in this particular point you can calculate b by solving the equation for b and then substituting x and y with one of your two points.

We can use the example above to illustrate this. We've got the two points (-3, 3) and (3, -1). From these two points we calculated the slope

$$m=-\frac{2}{3}$$

This gives us the equation

$$y=-\frac{2}{3}x+b$$

From this we can solve the equation for b

$$b=y+\frac{2}{3}x$$

And if we put in the values from our first point (-3, 3) we get

$$b=3+\frac{2}{3}\cdot \left ( -3 \right )=3+\left ( -2 \right )=1$$

If we put in this value for b in the equation we get

$$y=-\frac{2}{3}x+1$$

which is the same equation as we got when we read the y-intercept from the graph.

To summarize how to write a linear equation using the slope-interception form you

1. Identify the slope, m. This can be done by calculating the slope between two known points of the line using the slope formula.
2. Find the y-intercept. This can be done by substituting the slope and the coordinates of a point (x, y) on the line in the slope-intercept formula and then solve for b.

Once you've got both m and b you can just put them in the equation at their respective position.

### Video lesson

Find the equation to the graph

Sours: https://www.mathplanet.com/education/algebra-1/formulating-linear-equations/writing-linear-equations-using-the-slope-intercept-form

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T. R. Abdulmyanov

Abstract—The paper analyzes the origin for the separation of gas-and-dust disks around young stars into two large groups: those with circular and spiral structures. Such a separation of disk structures is explained by the existence of stable and unstable gas-dynamic equilibrium of protostellar envelopes. A model of wave disturbances of the gas-dust protostellar envelopes is constructed. The equation

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
J. Kotlarz; U. Zielenkiewicz; N. E. Zalewska; K. A. Kubiak

AbstractEnceladus is an attractive place to look for signs of life thanks to liquid water and the availability of energy. Recent research has proven that the ejected material of Enceladus south pole consists of water vapor, water ice, carbon dioxide, methane and molecular hydrogen. Possible similarities of physical and chemical conditions between Enceladus ocean bottom and the carbonate mineral matrix

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
N. A. Tikhonov; O. A. Galazutdinova; G. M. Karataeva

Abstract—Based on archived images from the Hubble Space Telescope, stellar photometry of the dwarf galaxy Coma P, located near the Virgo galaxy cluster, was performed. The selection of weakly luminous regions of the galaxy showed that Coma P consists of two interacting dwarf galaxies, one of which has a weak star formation, and the second galaxy has a very low surface brightness, there is no star formation

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
V. L. Plokhotnichenko

Abstract—This paper presents the functional design of “Quantochron 5-48” high temporal resolution system for multidimensional chronometration of stochastic fluxes. The device can determine the photon arrival times by array detectors and the accompanying spatial, spectral, and polarization characteristics of these photons, with a sampling time of 10 nanoseconds and the accuracy of UTC calibration provided

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
A. Kostenkov; A. Vinokurov; Y. Solovyeva; K. Atapin; S. Fabrika

AbstractWe calculate models of extended stellar atmospheres with a temperature in the range of 12 000–40 000 K and a mass loss rate of $${{10}^{{ - 6}}}$$–$${{10}^{{ - 4}}}{{M}_{ \odot }}$$$${{{\text{yr}}}^{{ - 1}}}$$. A large number of objects with emission spectra, such as luminous blue variables, FeII—emission line stars, Of/late-WN stars, and even ultraluminous X-ray sources often have effective

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
G. G. Valyavin; F. A. Musaev; A. V. Perkov; V. N. Aitov; V. D. Bychkov; S. V. Drabek; V. S. Shergin; D. A. Sazonenko; D. E. Kukushkin; G. A. Galazutdinov; E. V. Emelyanov; G. V. Yakopov; T. E. Burlakova; J.-L. Bertaux; A. V. Tavrov; O. I. Korablev; M. V. Yushkin; A. F. Valeev; D. R. Gadelshin; K.-M. Kim; Inwoo Han; B.-Ch. Lee

Abstract—We report the results of laboratory tests of the first version of the high-resolution fiber-fed spectrograph designed for the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS). During laboratory tests a spectral resolution of about 70 000 was achieved. The peak efficiency of the entire optical path of the spectrograph (without light losses at the CCD, primary mirror

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
A. V. Antipova; D. I. Makarov; L. N. Makarova

Abstract—We use the archival data obtained with the Hubble space telescope to measure distances by the tip of the red-giant branch (TRGB) method to SNIa parent galaxies. Our sample consists of four galaxies: NGC2841, NGC4496A, NGC4535, and NGC4527. For three of them, we were able to determine only the lower bounds of the distance. For the galaxy NGC4527, we estimated the TRGB distance modulus: (m − M)0

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-07-27
F. Buccella; G. V. Pallottino; P. Galeotti; G. Pizzella

AbstractThe correlation between the data obtained with two gravitational wave (GW) detectors (in Rome and in Maryland) and two neutrino detectors (LSD in Italy and Kamiokande in Japan) has been studied in more detail. If we indicate with U1 the response of the gravitational wave detectors and with U2 the response of the neutrino detectors, we find that there is a clear time correlation between U1 and

更新日期：2020-07-27
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
S. S. Kaisin; I. D. Karachentsev; H. Hernandez-Toledo; L. Gutierrez; V. E. Karachentseva

Abstract—We present the Hα images of ultra-flat (UF) spiral galaxies seen practically edgeon. The galaxies have the angular diameter in the B band $$a > 1.9'$$ and the apparent axial ratio (a/b) > 10. We found that their Hα images look, on average, almost two times thinner than those in the red continuum. The star-formation rate in the studied objects, determined from the Hα flux, is in good agreement

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
V. L. Plokhotnichenko; G. M. Beskin; S. V. Karpov; A. V. Solin; A. A. Solin; A. S. Terekhov; S. N. Kosolobov; G. E. Scheibler; V. G. de Bur; S. V. Moiseev; V. V. Pavlova; S. S. Moiseev

Abstract—We report the development of a photosensor device based on a position-sensitive detector with a gallium arsenide (GaAs) cathode and a 16-element anode. In the case of asymmetric heterostatic circuit (electrodes combined in fours) its working field was limited by 10 mm (for a 18-mm photocathode). Implementing a scheme of analog coding of the coordinates of the centroids of electron avalanches

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
V. A. Marsakov; V. V. Koval’; M. L. Gozha

Abstract—Using modern published data on velocities and spectroscopic definitions of chemical elements in stellar objects of the Galaxy, we investigated the relationship of chemical composition with the kinematics of different populations. The paper shows that the old stellar populations of the Galaxy, belonging (by the kinematic criterion) to the thick disk subsystem—globular clusters, field variables

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
V. L. Afanasiev; A. V. Moiseev; A. A. Smirnova

Abstract—We report the results of a detailed study of the central part of the Seyfert galaxy Mkn938. Observational data were obtained with the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences using integral field spectrograph MPFS and a scanning Fabry–Perot interferometer. Mkn938 is interesting for being a result of a merger of two gas-rich galaxies, and we

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
V. N. Chernenkov

Abstract—Requirements to a modern computer network of a research organization, in particular, to the local-area network of the Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) are described. The choice of the direction of the development of primary tools for supporting the observing process and based on local network infrastructure services of SAO RAS is defined. This

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
L. V. Yasnov; V. M. Bogod; A G. Stupishin

Abstract—We present a method of determination of numbers of gyroresonance layers most effectively radiating in the transition region of the active solar atmosphere. It is based on determination of frequency in the spectrum of an extraordinary wave of an active region, at which the gradient of this spectrum abruptly rises. Observations of 29 active regions carried out at the RATAN-600 radio telescope

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2020-04-15
A. V. Ivanova

Abstract—A detailed study of comets active at large heliocentric distances (greater than 4 au) which enter the Solar System for the first time and are composed of matter in its elementary, unprocessed state, would help in our understanding of the history and evolution of the Solar System. In particular, contemporary giant planet formation models require the presence of accretion of volatile elements

更新日期：2020-04-15
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
R. I. Uklein; E. A. Malygin; E. S. Shablovinskaya; A. E. Perepelitsyn; A. A. Grokhovskaya

The improvement of the calibration relation for determining the size of the broad-line region from the observed optical luminosity of active galactic nuclei (AGN) is a necessary task to study fundamental parameters of distant AGNs such as mass of the central supermassive black hole. The most popular method of the BLRsize estimation is the reverberation mapping based onmeasuring the time delay between

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
V. V. Pushkarev; E. K. Majorova; O. V. Verkhodanov

The spectral properties of inhomogeneities of the cosmic microwave background are considered. The inhomogeneity spectra are studied in the frequency range from 30 to 217 GHz according to multi-frequency maps of the Planck space mission with smoothing at different angular sizes. According to the thermodynamic temperature versus frequency relation, it is shown that the map at 70 GHz has a signal distribution

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
N. A. Tikhonov

We conducted the stellar photometry using the archive images of the Hubble Space Telescope of the dwarf irregular galaxy Cas 1 located near the Milky Way. Blue supergiants detected in the obtained Hertzsprung-Russell diagram are concentrated in star-forming regions. We found that the red star branch consists of numerous AGB stars and a small number of brighter red supergiants. In the luminosity function

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
Yu. V. Sotnikova; T. V. Mufakharov; E. K. Majorova; M. G. Mingaliev; R. Yu. Udovitskii; N. N. Bursov; T. A. Semenova

Gigahertz-Peaked spectrum (GPS) sources are compact active galactic nuclei, presumably young precursors of bright radio sources. The study of GPS radio properties provides information about the features of synchrotron radiation in extragalactic sources. Also in applied research, GPS sources are useful as compact stationary radio sources in the sky for astrometric purposes. This paper presents the results

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
V. A. Marsakov; V. V. Koval’; M. L. Gozha

A comprehensive statistical analysis of the relationship between the chemical and spatially kinematic parameters of the globular clusters of the Galaxy has been performed. The data of the author’s compilation catalog contain astrophysical parameters for 157 clusters and the relative abundances of α-elements for 69 clusters. For 121 clusters, the data are supplemented by spatially kinematic parameters

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
F. G. Kopylova; A. I. Kopylov

We study the nearest outskirts (R < 3R200c) of 40 groups and clusters of galaxies of the local Universe (0.02 < z < 0.045 and 300 km s-1 < σ < 950 km s-1). Using the SDSS DR10 catalog data, we determined the stellar mass of galaxy clusters corresponding to Ks-luminosity (which we determined earlier based on the 2MASX catalog data) (M*/M⊙) ∝ (LK/L⊙)1.010±0.004 (MK < - 21m.5, R < R200c). We also found

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
V. S. Kozyreva; A. V. Kusakin; T. Krajci; A. I. Bogomazov

The paper describes a target setting of searching for apsidal motion in 21 eclipsing binaries. We estimate the capability of its determination for the systems of interest in the coming years, as well as present results of observations in 2009–2018, in which the minima of the light curves of these stars have been obtained. The accuracy of orbital period determination of the studied binaries is 10–100

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
A. A. Grokhovskaya; S. N. Dodonov

We present the results of methodological works on automated analysis of the large scale distribution of galaxies. Selecting candidates for clusters and groups of galaxies was carried out using two complementary methods of determining the density contrast maps in the narrow layers of the three-dimensional large scale distribution of galaxies: the filtering algorithm with an adaptive core and the Voronoi

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
V. Šimon; H. Edelmann

We analyze the long-term activity of a little known dwarf nova (DN) DT Oct. We use photographic observations from the Bamberg Observatory (Southern Patrol Photographic Sky Survey), CCD data from ASAS-3 survey, and AAVSO data (both CCD and visual). We determined the recurrence times of the superoutbursts, TCso, and of the normal outbursts, TCno, to be approximately 311 d and about 30 d, respectively

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
A. I. Kolbin; V. V. Shimansky; I. Yu. Alekseev; N. N. Shimanskaya; M. M. Gabdeev; V. V. Tsymbal

We present an analysis of the spectra of the star V837 Tau, which belongs to a class of chromosphere-active RSCVn type stars. The observational material was obtained with the 1.5-m Russian-Turkish telescope RTT-150 and the 2.6-m Shajn telescope of the Crimean astrophysical observatory. To reveal signs of the photosphere spottedness of the primary and to isolate the faint secondary, we performed the

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
I. S. Savanov

We study relations of the parameter Rvar (amplitude of the brightness variability, stellar activity indicator) with respect to the effective temperature and the period of rotation of stars belonging to the A–M spectral types. A common dataset gives an idea of variations in this parameter for stars with convection. It was confirmed that the limit of applicability for the gyrochronological relation PtM

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
S. G. Masda; J. A. Docobo; A. M. Hussein; M. K. Mardini; H. A. Al-Ameryeen; P. P. Campo; A. R. Khan; J. M. Pathan

The precise determination of the physical and dynamical parameters of the HIP 109951 triple star system (WDS J22161-0705AB) which is formed by the A, Ba, and Bb components are presented. The binary nature of component B was recently confirmed by studying the radial velocities. The analysis of the system follows Al-Wardat's complex method for analyzing CVBS which employs Kurucz (Atlas9) line blanketed

更新日期：2019-12-10
• Astrophys. Bull. (IF 1.191) Pub Date : 2019-12-10
V. G. Klochkova

We present the results of long-term spectral monitoring of yellow hypergiants (YHGs) of northern hemisphere with a R ≥ 60 000 resolution. The spectra of these F–G stars of extreme luminosity, compactly located at the top of the Hertzsprung–Russell diagram revealed a variety of spectral features: various types of Hα profile, the presence (or absence) of forbidden and permitted emissions, as well as

更新日期：2019-12-10
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