Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables.
Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables. Represent the Cartesian coordinate system and identify the origin and axes.
Given an ordered pair, locate that point on the Cartesian coordinate system. Given a point on the Cartesian coordinate system, state the ordered pair associated with it.
We have already used the number line on which we have represented numbers as points on a line. Note that this concept contains elements from two fields of mathematics, the line from geometry and the numbers from algebra.
Rene Descartes devised a method of relating points on a plane to algebraic numbers. This scheme is called the Cartesian coordinate system for Descartes and is sometimes referred to as the rectangular coordinate system.
This system is composed of two number lines that are perpendicular at their zero points. Perpendicular means that two lines are at right angles to each other. Study the diagram carefully as you note each of the following facts.
The number lines are called axes. The horizontal line is the x-axis and the vertical is the y-axis. The zero point at which they are perpendicular is called the origin.
Positive is to the right and up; negative is to the left and down. The arrows indicate the number lines extend indefinitely. Thus the plane extends indefinitely in all directions. The plane is divided into four parts called quadrants. These are numbered in a counterclockwise direction starting at the upper right.
Points on the plane are designated by ordered pairs of numbers written in parentheses with a comma between them, such as 5,7. This is called an ordered pair because the order in which the numbers are written is important. The ordered pair 5,7 is not the same as the ordered pair 7,5.
Points are located on the plane in the following manner. First, start at the origin and count left or right the number of spaces designated by the first number of the ordered pair.
Second, from the point on the x-axis given by the first number count up or down the number of spaces designated by the second number of the ordered pair. Ordered pairs are always written with x first and then y, x,y.
The numbers represented by x and y are called the coordinates of the point x,y. The first number of the ordered pair always refers to the horizontal direction and the second number always refers to the vertical direction.
Check each one to determine how they are located. What are the coordinates of the origin? Find several ordered pairs that make a given linear equation true.
Locate these points on the Cartesian coordinate system. Draw a straight line through those points that represent the graph of this equation.
A graph is a pictorial representation of numbered facts. There are many types of graphs, such as bar graphs, circular graphs, line graphs, and so on. You can usually find examples of these graphs in the financial section of a newspaper.
Graphs are used because a picture usually makes the number facts more easily understood. In this section we will discuss the method of graphing an equation in two variables.
In other words, we will sketch a picture of an equation in two variables. All possible answers to this equation, located as points on the plane, will give us the graph or picture of the equation. A sketch can be described as the "curve of best fit.Write an equation in slope-intercept form for the line that passes through (1, 7) and (5, 3).
y = - x + 8 A linear function has the same y-intercept as x + 4y = 12 and its graph contains the point (5, 4).
Feb 02, · Grab a pencil and paper and study along with me! In this video, you will be given the graph of a linear inequality and write the inequality based on what you see. You can write an inequality to represent a situation. Write Inequalities with Write an inequality for each sentence.
You must be over 12 years old to ride the go-karts. Words Variable Inequality Your age is over Let a = your age. a > 12 The inequality is a > A pony is less than hands tall. Words Variable Inequality A pony is less than To solve a system of two linear inequalities by graphing, determine the region of the plane that satisfies both inequality statements.
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To graph inequalities, use graphing calculator. GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES () Graph of a Linear Inequality The graph of a linear inequality in two variables consists of all points in the rectan- Write an inequality that limits the possi-ble number of tables of each type that can be made.
Draw a graph showing all.