Linear Equations and Word Problems
We will define the term variable which is analogous to a pronoun in grammar.
is a symbol that represents a number
that may change depending on the circumstance.
We often use the variable t to represent time. We can
use the variable h to represent height. We use the letter
x for the
variable when the variable stands for a general number without context.
is a symbol that represents a known fixed
c = 3 x 108
An algebraic expression
is an expression
consisting of constants, variables, and symbols of operators.
3x2 +2x - 1
is an expression that is related by addition or subtraction.
3x2 +2x -
are the three terms. 3, 2,
-1 are called
the coefficients of
x2, x, and
1 respectively. We say two
terms are like terms if they differ only in their coefficient.
are like terms. We can add like terms by adding their coefficients.
3x2 + 5x2
Simplify the expression
- 2p + 3x2 + 5s2 + 2p -
Recall that an expression is a statement involving x's, constants, and operators.
An expression is like the subject of a sentence. An equation
is more like a complete sentence which as a subject, verb and predicate.
The verb will typically be an "= " sign.
3x + 1 = 4
is an example of an equation.
When we have an equation, we can attempt to solve it that is to find which x values
satisfy the equation. An x value satisfies an equation if when we
substitute the value in the equation for x, the equation is true. To solve for the variable
x means that we have an equation of the form.
x = a number.
The set of all x that satisfy the equation is called the solution
set of the equation.
To solve we isolate x by
Use the distributive property.
Combine like terms.
Subtract all the x terms from the right hand side- "Do
unto the left as you would do unto the right."
Subtract all the constant terms from the left hand side.
Divide by the coefficient in front of x.
Solve the following equation for x.
3(2x - 1) + 4x =
2(4x + 2)
6x - 3 + 4x = 8x + 4
10x - 3 = 8x + 4
2x - 3 = 4
2x = 7
x = 7/2
In the above example
we say that
x = 7/2
is a solution to the equation.
3(2x - 1) - 2(3x + 1) = 0
6x - 3 - 6x - 2
= 0 (Distributed the
3 and the 2)
0 - 5 = 0
-5 = 0
Since this can never happen we say that the equation is a contradiction.
2(x - 3) - 2x + 8 = 2
2x - 6 - 2x + 8
= 2 (Distributed the
0 + 2 = 2
2x - 2x and the
-6 + 8)
Since this is always true, we say that the equation is an identity.
Problem Solving With Formulas
A formula is an equation that relates real world quantities.
2l + 2w
is the formula for the perimeter P of a rectangle given the length
l and width w.
is the formula for the distance traveled d given the speed
and the time t.
P + Prt
is the formula for the amount A in a bank account
t years after
P dollars is put in at an interest rate of r.
V = pr2h
is the formula for the volume V of a cylinder of radius
r and height h, where p @ 3.14.
(b1 + b2)
is the formula for the area A of
a trapazoid with height h and bases b1 and
We say that a formula is solved for a variable x if the equation becomes
where the left hand side does not include any x's.
C = 2pr
for r then determine the radius of a circle with circumference
Divide both sides by 2p:
Use the reflexive property to get
Now plug in 4 for C to obtain
Steps for Solving a Word Problem
Read the problem, sketch the proper picture, and label variables.
Write down what the answer should look like.
Come up with the appropriate formula.
Solve for the needed variable.
Plug in the known numbers.
Answer the question.
A pile of sand has the shape of a right circular cone.
Find the height of the pile if it contains 100 cc of sand and the radius
is 5 cm.
The height of the pile is ________ cm.
We use the formula for the volume of a right circular cone:
Multiply by 3 on both sides to get
Divide both sides by pr2 to obtain
= h or
The height of the pile is 12/p cm.
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