Intermediate Algebra (11th edition)

NOW TRY

We can find yby substituting for zin equation (2).

(2)

Let

Add 6.

or Divide by 8; lowest terms

Check to verify that the solution set is NOW TRY

OBJECTIVE 4 Solve special systems.

Solving an Inconsistent System with Three Variables

Solve the system.

(1)

(2)

x - 2 y+ 3 z = 1 (3)

- x+ 3 y- 2 z =- 1

2 x- 4 y+ 6 z = 5

EXAMPLE 3

EA

2

3 ,

3

4 , -^6 BF.

3

4

y=

6

8

,

8 y= 6

8 y- 6 = 0 z=-6.

8 y+z= 0

- 6

230 CHAPTER 4 Systems of Linear Equations

NOW TRY
EXERCISE 3
Solve the system.

• 2 x+ 10 y- 4 z= 7

3 x+ y- z= 6

x- 5 y+ 2 z= 4

Eliminate the focus variable,x, using equations (1) and (3).

Multiply each side of (3) by.

(1)

Add; false

The resulting false statement indicates that equations (1) and (3) have no common

solution. Thus, the system is inconsistent and the solution set is The graph of this

system would show the two planes parallel to one another.

0.

0 = 3

2 x-^4 y+^6 z=^5

- 2 x+ 4 y- 6 z=- 2 - 2

Use as the working

equation, with

focus variable x.

NOW TRY

NOTE If a false statement results when adding as in Example 3,it is not necessary

to go any further with the solution. Since two of the three planes are parallel, it is not

possible for the three planes to have any points in common.

Solving a System of Dependent Equations with Three Variables

Solve the system.

(1)

(2)

(3)

Multiplying each side of equation (1) by 3 gives equation (3). Multiplying each

side of equation (2) by also gives equation (3). Because of this, the equations are

dependent. All three equations have the same graph, as illustrated in FIGURE 7(c). The

solution set is written as follows.

Set-builder notation

Although any one of the three equations could be used to write the solution set, we

use the equation in standard form with coefficients that are integers with greatest

common factor 1, as we did in Section 4.1.

51 x, y, z 2 | 2 x- 3 y+ 4 z= 86

- 6

6 x- 9 y+ 12 z= 24

- x+

3

2

y- 2 z=- 4

2 x- 3 y+ 4 z= 8

NOW TRY EXAMPLE 4

EXERCISE 4
Solve the system.

1

2

x-

3

2

y+ z= 5

• 2 x+ 6 y- 4 z=- 20

x- 3 y+ 2 z= 10

NOW TRY ANSWERS
2.
3.

4. 51 x, y, z 2 | x- 3 y+ 2 z= 106

0

51 - 2, 1, 4 26

NOW TRY
EXERCISE 2
Solve the system.

x- 6 y=- 8

4 y+ 5 z= 24

3 x- z=- 10