Beginning Algebra, 11th Edition

Identifying the Three Cases by Using Slopes

Describe each system without graphing. State the number of solutions.

(a)

Write each equation in slope-intercept form, by solving for y.

Subtract 3x. Divide by.

Divide by 2.

Both equations have slope but they have different y-intercepts, 3 and In

Section 3.3,we found that lines with the same slope are parallel, so these equations

have graphs that are parallel lines. Thus, the system has no solution.

(b)

Again, write the equations in slope-intercept form.

Subtract 2x. Interchange sides.

Multiply by. Subtract 2.

Multiply by 2.

The equations are exactly the same—their graphs are the same line. Thus, the system

has an infinite number of solutions.

(c)

In slope-intercept form, the equations are as follows.

Subtract x. Subtract 2x.

Divide by.

The graphs of these equations are neither parallel nor the same line, since the slopes

are different. This system has exactly one solution. NOW TRY

NOTE The solution set of the system in Example 4(a)is , since the graphs of the

equations of the system are parallel lines. The solution set of the system in Exam-

ple 4(b),written using set-builder notation and the first equation, is

If we try to solve the system in Example 4(c)by graphing, we will have difficulty

identifying the point of intersection of the graphs. We introduce an algebraic method

for solving systems like this in Section 4.2.

51 x, y 2 | 2 x- y= 46.

0

y= - 3

1

3

x-

5

3

- 3 y=-x+ 5 y = - 2 x+ 8

x- 3 y= 5 2 x+y= 8

2 x+ y= 8

x- 3 y= 5

y= 2 x- 4

y

2

y= 2 x- 4 - 1 =x- 2

y

2

- y=- 2 x+ 4 + 2 =x

x=

y

2

2 x- y= 4 + 2

x=

y

2

+ 2

2 x- y= 4

5

- 2.

3 2

y= -

3

2

x+ 3

y=- - 2

3

2

x+

5

2

2 y=- 3 x+ 6

3 x+ 2 y= 6 - 2 y= 3 x- 5

y= mx+b,

- 2 y= 3 x- 5

3 x+ 2 y= 6

EXAMPLE 4

252 CHAPTER 4 Systems of Linear Equations and Inequalities

NOW TRY
EXERCISE 4
Describe each system without
graphing. State the number of
solutions.

(a)

(b)

(c)

3 y- x= 0

y- 3 x= 7

3 y=- 6 x- 12

2 x+y= 7

x-^85 y=^45

5 x- 8 y= 4

NOW TRY ANSWERS

(a)The equations represent the
same line. The system has an
infinite number of solutions.
(b)The equations represent
parallel lines. The system
has no solution.
(c)The equations represent lines
that are neither parallel nor
the same line. The system
has exactly one solution.

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