Precalculus, Chapter 7, 7.3, Section 7.3, Problem 45

3 w+x = 4 (equation 1)
-w+2 y-z = 0 (equation 2)
3 y-2 w = 1 (equation 3)
2 x-y+4 z = 5 (equation 4)
Add 1/3 × (equation 1) to equation 2:3 w+x+0 y+0 z = 4 (equation 1)
0 w+x/3+2 y-z = 4/3 (equation 2)
-(2 w)+0 x+3 y+0 z = 1 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Multiply equation 2 by 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+x+6 y-3 z = 4 (equation 2)
-(2 w)+0 x+3 y+0 z = 1 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Add 2/3 × (equation 1) to equation 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+x+6 y-3 z = 4 (equation 2)
0 w+(2 x)/3+3 y+0 z = 11/3 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Multiply equation 3 by 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+x+6 y-3 z = 4 (equation 2)
0 w+2 x+9 y+0 z = 11 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Swap equation 2 with equation 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+x+6 y-3 z = 4 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Subtract 1/2 × (equation 2) from equation 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x+(3 y)/2-3 z = (-3)/2 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Multiply equation 3 by 2/3: 3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x+y-2 z = -1 (equation 3)
0 w+2 x-y+4 z = 5 (equation 4)
Subtract equation 2 from equation 4:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x+y-2 z = -1 (equation 3)
0 w+0 x-10 y+4 z = -6 (equation 4)
Divide equation 4 by 2:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x+y-2 z = -1 (equation 3)
0 w+0 x-5 y+2 z = -3 (equation 4)
Swap equation 3 with equation 4:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x-5 y+2 z = -3 (equation 3)
0 w+0 x+y-2 z = -1 (equation 4)
Add 1/5 × (equation 3) to equation 4:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x-5 y+2 z = -3 (equation 3)
0 w+0 x+0 y-(8 z)/5 = (-8)/5 (equation 4)
Multiply equation 4 by -5/8:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x-5 y+2 z = -3 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Subtract 2 × (equation 4) from equation 3:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x-5 y+0 z = -5 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Divide equation 3 by -5:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+9 y+0 z = 11 (equation 2)
0 w+0 x+y+0 z = 1 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Subtract 9 × (equation 3) from equation 2:3 w+x+0 y+0 z = 4 (equation 1)
0 w+2 x+0 y+0 z = 2 (equation 2)
0 w+0 x+y+0 z = 1 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Divide equation 2 by 2:3 w+x+0 y+0 z = 4 (equation 1)
0 w+x+0 y+0 z = 1 (equation 2)
0 w+0 x+y+0 z = 1 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Subtract equation 2 from equation 1:3 w+0 x+0 y+0 z = 3 (equation 1)
0 w+x+0 y+0 z = 1 (equation 2)
0 w+0 x+y+0 z = 1 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)
Divide equation 1 by 3:w+0 x+0 y+0 z = 1 (equation 1)
0 w+x+0 y+0 z = 1 (equation 2)
0 w+0 x+y+0 z = 1 (equation 3)
0 w+0 x+0 y+z = 1 (equation 4)


w = 1
x = 1
y = 1
z = 1