Intermediate Algebra (11th edition)

OBJECTIVE 5 Solve special systems.As we saw in FIGURES 3(b) AND (c), some

systems of linear equations have no solution or an infinite number of solutions.

Solving a System of Dependent Equations

Solve the system.

(1)

(2)

We multiply equation (1) by and then add the result to equation (2).

times each side of equation (1)

(2)

True

Adding these equations gives the true statement

In the original system, we could get equation (2) from

equation (1) by multiplying equation (1) by 3. Because

of this, equations (1) and (2) are equivalent and have the

same graph, as shown in FIGURE 4. The equations are

dependent.

The solution set is the set of all points on the line

with equation written in set-builder nota-

tion (Section 1.1)as

and read “the set of all ordered pairs such that ” NOW TRY

NOTE When a system has dependent equations and an infinite number of solutions,

as in Example 8,either equation of the system could be used to write the solution set.

In this book, we use the equation in standard form with coefficients that are inte-

gers having greatest common factor 1 and positive coefficient of x.

Solving an Inconsistent System

Solve the system.

(1)

(2)

Multiply equation (1) by 2, and then add the result to equation (2).

Equation (1) multiplied by 2

(2)

False

The result of the addition step is a false statement, which indicates that the system

is inconsistent. As shown in FIGURE 5, the graphs of the equations of the system are

parallel lines.

There are no ordered pairs that satisfy both equations, so there is no solution for

the system. The solution set is 0.

0 = 11

-^2 x-^6 y=^3

2 x+ 6 y= 8

- 2 x- 6 y= 3

x+ 3 y= 4

EXAMPLE 9

1 x, y 2 , 2 x- y=3.

51 x, y 2 | 2 x-y= 36

2 x-y= 3,

0 = 0.

0 = 0

6 x-^3 y=^9

- 6 x+ 3 y=- 9

- 3

6 x- 3 y= 9

2 x- y= 3

EXAMPLE 8

SECTION 4.1 Systems of Linear Equations in Two Variables 217

NOW TRY
EXERCISE 8
Solve the system.

• 3 x+ 9 y=- 21

x- 3 y= 7

x

y

02

–3

Same line —

inf initely

many

solutions

2 x – y = 3

6 x – 3y = 9

FIGURE 4

NOW TRY
EXERCISE 9
Solve the system.

6 x- 15 y= 4

• 2 x+ 5 y= 6

NOW TRY ANSWERS
8.

9. 
   

51 x, y 2 |x- 3 y= 76

NOW TRY

x

y

0

–1 4

Parallel

lines —

no solution

x + 3y = 4

–2x – 6y = 3

FIGURE 5