3 Graphs of Linear Equations, and Inequalities, in Two ...

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Graphs of Linear Equations, and Inequalities, in Two Variables

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3.1 Linear Equations in Two Variables; The Rectangular Coordinate System

Objectives

1. Interpret graphs. 2. Write a solution as an ordered pair. 3. Decide whether a given ordered pair is a

solution of a given equation. 4. Complete ordered pairs for a given

equation. 5. Complete a table of values. 6. Plot ordered pairs.

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Interpret Graphs

Example 2

(a) Which quiz had the highest class

The line graph below

average? Quiz #3

shows the class averages on the first eight quizzes in a college math course. 1010

(b) Which quiz experienced the biggest jump in class average from the previous quiz? Quiz #5

(c) Estimate the difference

7.5

between the class average

Score 5

on Quiz #2 and Quiz #3. Difference 9 ? 7

2.5

0 Quiz #1 Quiz #2 Quiz #3 Quiz #4 Quiz #5 Quiz #6 Quiz #7 Quiz #8

Difference 2

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Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form

Ax + By = C, where A, B, and C are real numbers and A and B are not both 0.

Note Other linear equations in two variables, such as

y = 4x + 5 and 3x = 7 ? 2y, are not written in standard form but could be. We discuss the forms of linear equations in Section 3.4.

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Write a Solution as an Ordered Pair

A solution of a linear equation in two variables requires two numbers, one for each variable. For example, a true statement results when we replace x with 2 and y with 13 in the equation y = 4x + 5 since

13 = 4(2) + 5.

Let x = 2, y = 13.

The pair of numbers x = 2 and y = 13 gives one solution of the equation y = 4x + 5. The phrase "x = 2 and y = 13" is abbreviated

x-value

y-value

(2,13)

Ordered Pair

The x-value is always given first. A pair of numbers such as (2,13) is called an ordered pair.

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Decide Whether an Ordered Pair is a Solution Example 3

Decide whether each ordered pair is a solution to 5x ? 2y = 4.

(a) (2,3)

To see whether (2,3) is a solution, substitute 2 for x and 3 for y. 5(2) ? 2(3) =? 4

10 ? 6 =? 4

4 = 4 True

Thus, (2,3) is a solution.

(b) (?2,?3)

To see whether (?2,?3) is a solution, substitute ?2 for x and ?3 for y. 5(?2) ? 2(?3) =? 4

?10 + 6 =? 4

?4 = 4 False

Thus, (?2,?3) is not a solution.

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Complete Ordered Pairs

Example 4 Complete each ordered pair for the equation ?3x + y = 4.

(a) (3, ) Substitute 3 for x and solve for y.

?3(3) + y = 4

?9 + y = 4 +9 +9

y = 13

The ordered pair is (3,13).

(b) ( ,1)

Substitute 1 for y and solve for x.

?3x + 1 = 4 ?1 ?1

?3x = 3 3 3

x = ?1 The ordered pair is (?1,1).

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Complete a Table of Values

Example 5 Complete the table of values for each equation.

(a) x ? 3y = 6

xy ?1

12

x ?3(?1) = 6

x +3 = 6 ?3 ?3 x=3

12 ? 3y = 6 ? 12 ? 12

? 3y = ?6 3 3

y=2

For the first ordered pair, let y = ?1.

For the second ordered pair, let x = 12.

xy Thus, the completed 3 ?1 table is: 12 2

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