Copyright © 2013 Pearson Education, Inc. Section 3.2 Linear Equations in Two Variables.

Copyright © 2013 Pearson Education, Inc.

Section 3.2

Linear Equations in Two Variables

Example

Determine whether the given ordered pair is a solution to the given equation.a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)

Solution

a. y = x + 57 = 2 + 57 = 7 True

The ordered pair (2, 7) is a solution.

Page 169

Example

Determine whether the given ordered pair is a solution to the given equation.a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)

Solution

b. 2x + 3y = 18

2(3) + 3(4) = 18

6 12 = 18

6 18

The ordered pair (3, 4) is NOT a solution.

Page 169

ExampleA table can be used to list solutions to an equation.

Complete the table for the equation y = 3x – 1.

Solution

x 3 1 0 3

y

3x

3 1

3( 3) 1

9 1

10

y x

y

y

y

10

1x

3 1

3( ) 1

3

4

1

1

y x

y

y

y

0x

3 1

3( ) 1

0

1

0

1

y x

y

y

y

3x

3 1

3( ) 1

9

8

3

1

y x

y

y

y

4 1 8

Page 170

Find five solutions to the equation y = 3x + 2.

Choose some x values and then compute the corresponding y values and complete the table.

If x = -2, y = 3(-2) + 2 = -4. Ordered pair

If x = -1, y = 3(-1) + 2 = -1. Ordered pair

If x =0, y = 3(0) + 2 = 2. Ordered pair



Graph on next slide.

Finding Solutions of an Equation

X Y (x,y)

-2

-1

0

1

2

Page 170

)4,2(4

)1,1(1

)2,0(2

)5,1(5

)8,2(8

A table that lists a few solutions is helpful when graphing an equation

Plot the five ordered pairs to obtain the graph of y = 3x + 2

Graph of the Equation

X Y (X,Y)

-2 -4 (-2,-4)

-1 -1 (-1,-1)

0 2 (0,2)

1 5 (1,5)

2 8 (2,8)

Example

Make a table of values for the equation y = 3x, and then use the table to graph this equation.SolutionStart by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table. x y

–1 –3

0 0

1 3

2 6Plot the points and connect the points with a straight line.

Page 171

A linear equation in two variables can be written as

Ax + By = C,

where A, B, and C are fixed numbers (constants) and A and B are not both equal to 0.

The graph of a linear equation in two variables is a line.

Linear Equation in Two VariablesPage 172

Example

Graph the linear equation. SolutionBecause this equation can be written in standard form, it is a linear equation.

Choose any three values for x.

11

4y x

x y

–4 0

0 1

4 2

Plot the points and connect the points with a straight line.

Page 173

Example

Graph the linear equation. SolutionBecause this equation can be written in standard form, it is a linear equation.

Choose any three values for x.

Plot the points and connect the points with a straight line.

5x y

x y

0 5

2 3

5 0

Page 173

Example

Graph the linear equation by solving for y first.

SolutionSolve for y.

3 6 12x y

3 6 12x y

6 3 12y x 1

22

y x

x y

–2 1

0 2

2 3

Page 174


(2,7)

(1,4)

(0,1)

(-1,-2)

(-2,-5)


X Y (x,y)

2 7 (2,7)

1 4 (1,4)

0 1 (0,1)

-1 -2 (-1,-2)

-2 -5 (-2,-5)

Tables of Solutions

A table can be used to list solutions to an equation.

A table that lists a few solutions is helpful when graphing an equation.

Page 170

Basic Concepts

Equations can have any number of variables.

A solution to an equation with one variable is one number that makes the statement true.

8

Example

Graph the linear equation by solving for y first.

SolutionSolve for y.

3 6 12x y

3 6 12x y

6 3 12y x 1

22

y x

x y

–2 1

0 2

2 3

Page 174

Plot the ordered pairs to obtain the graph of y = 2x


2

)1(2

y

y

0

)0(2

y

y

4

)2(2

y

y

4

)2(2

y

y

Plot the ordered pairs to obtain the graph of y = 2x-2


0

2)1(2

y

y

2

2)0(2

y

y

2

2)2(2

y

y

6

2)2(2

y

y

Plot the ordered pairs to obtain the graph of


22

1 xy

3

2)2(2

1

y

y

2

2)0(2

1

y

y

4

2)4(2

1

y

y

1

2)2(2

1

y

y



X Y (x,y)

-2 -4 (-2,-4)

-1 -1 (-1,-1)

0 2 (0,2)

1 5 (1,5)

2 8 (2,8)

Copyright © 2013 Pearson Education, Inc. Section 3.2 Linear Equations in Two Variables.

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