Intermediate Algebra (11th edition)

The results of Examples 8 and 9are generalized as follows.

218 CHAPTER 4 Systems of Linear Equations

NOW TRY
EXERCISE 10
Write each equation in slope-
intercept form and then tell
how many solutions the
system has.

(a)

(b)

• 10 y= 2 x+ 8

5 y=-x- 4

4 x- 6 y=- 6

2 x- 3 y= (^3) (2)

2 x- y= 3 Divide by 3.

6 x- 3 y= 9

This leads to the

same result as

on the left.

The lines have the same slope and same y-intercept, meaning that they coincide.

There are infinitely many solutions.

Solve each equation from Example 9for y.

(1)

Subtract x.

Divide by 3.

Slope y-interceptA0,^43 B

y=-

1

3

x+

4

3

3 y=-x+ 4

x+ 3 y= 4 (2)

Add 2x.

Divide by.

Slope y-interceptA0, -^12 B

y=- - 6

1

3

x-

1

2

- 6 y= 2 x+ 3

- 2 x- 6 y= 3

The lines have the same slope, but different y-intercepts, indicating that they are

parallel. Thus, the system has no solution.

Slopes and y-intercepts can be used to decide whether the graphs of a system of

equations are parallel lines or whether they coincide.

Using Slope-Intercept Form to Determine the Number

of Solutions

Refer to Examples 8 and 9.Write each pair of equations in slope-intercept form, and

use the results to tell how many solutions the system has.

Solve each equation from Example 8for y.

(1)

Subtract 2x.

Multiply by.

Slope y-intercept 1 0, - 32

y= 2 x- 3 - 1

- y=- 2 x+ 3

2 x- y= 3

EXAMPLE 10

Special Cases of Linear Systems

If both variables are eliminated when a system of linear equations is solved, then

the solution sets are determined as follows.

Case 1 There are infinitely many solutions if the resulting statement is true.

Case 2 There is no solution if the resulting statement is false.

In Example 2,we showed how to solve the system

(1)

(2)

by graphing the two lines and finding their point of intersection. We can also do this

with a graphing calculator, as shown in FIGURE 6. The two lines were graphed by solv-

ing each equation for y.

Equation (1) solved for y

y= 2 x- 4 Equation (2) solved for y

y= 5 - x

2 x- y= 4

x+y= 5

CONNECTIONS

FIGURE 6

NOW TRY

NOW TRY ANSWERS

10. (a) ;
    no solution
    (b)Both are ;
    infinitely many solutions

y=-^15 x-^45

y= 32 x-1;y= 32 x+ 1