SECTION 4.2 Systems of Linear Equations in Three Variables 231
Solving Another Special SystemSolve the system.
(1)(2)(3)Multiplying each side of equation (2) by 2 gives equation (1), so these two equa-
tions are dependent. Equations (1) and (3) are not equivalent, however. Multiplying
equation (3) by does notgive equation (1). Instead, we obtain two equations with
the same coefficients, but with different constant terms.
The graphs of equations (1) and (3) have no points in common (that is, the planes
are parallel). Thus, the system is inconsistent and the solution set is as illustrated in
FIGURE 7(g).
0 ,
1
24 x- 2 y+ 6 z = 1
x-
1
2
y+
3
2
z= 3
2 x- y+ 3 z= 6
EXAMPLE 5
NOW TRYNOW TRY
EXERCISE 5
Solve the system.
1
2
x-3
2
y+ z= 21
3
x- y+2
3
z= 7x- 3 y+ 2 z= 49.
x- y-z=- 22 x+ y-z=- 1x+ 2 y+z= 4Complete solution available
on the Video Resources on DVD
4.2 EXERCISES
1.Concept Check The two equations have a common solution of. Which equation would complete a system of three linear equations in three vari-
ables having solution set?
A. B.
C. D.
2.Complete the work of Example 1and show that the ordered triple is also a
solution of equations (2) and (3).
Equation (2)
Equation (3)
Solve each system of equations. See Example 1.2 x- 3 y+ 2 z= 3x+ 7 y- 3 z=- 141 - 3, 1, 6 2
3 x+ 2 y-z= 5 3 x+ 2 y-z= 63 x+ 2 y-z= 1 3 x+ 2 y-z= 451 1, 2, 3 26
1 1, 2, 3 2
x+y+z= 6
2 x-y+z= 33.
x- 2 y- 4 z=- 5x+ 4 y- 2 z= 92 x- 5 y+ 3 z=- 1 4.x+ 2 y+ 2 z=- 12 x- y+ z= 1x+ 3 y- 6 z= 7 5.x+ 4 y- z= 202 x- 3 y+ 2 z=- 163 x+ 2 y+ z= 86.
2 x+ 3 y- 2 z=- 5- 4 x+ 2 y+ 3 z=- 1
- 3 x+ y- z=- 10 7.
3 x- 8 y- 2 z=- 64 x- 7 y- 3 z= 12 x+ 5 y+ 2 z= 0 8.2 x+ 4 y- 2 z= 144 x+ 3 y+ 5 z= 45 x- 2 y+ 3 z=- 910.
- 3 x+ 5 y- z=- 7
- 2 x- 3 y+ 4 z=- 14
x- 2 y+ 5 z=- 712.
x- y- z= 0x+ 2 y+ z= 22 x+ y+ 2 z= 111.
x+ 4 y- 3 z= 13 x+ 2 y+ 6 z= 6- x+ 2 y+ 6 z= 2
13.
- x+ 2 y- 3 z=- 4
2 x- y+ z=- 5x+ y- z=- 2 14.- 6 x+ y+ z=- 2
- x- y+ 3 z= 2
x+ 2 y+ 3 z= 1NOW TRY ANSWER