System of Equations with 3 variables.

Question

Solve the System of Equations

4x-y+3z=12

2x+9z=-5

x+4y+6z=-32

Answer 1

x + 4y + 6z = -32

2x + 9z = -5

4x - y + 3z = 12

Add -2 times the first equation to the second and -4 times the first equation to the third:

x + 4y + 6z = -32

- 8y - 3z = 59

-17y -21z = 140

Add 2 times the second equation to the first and -7 times the second equation to the third:

x - 12y = 86

- 8y - 3z = 59

39y = -273

So, y = -7.

From the first equation, x + 84 = 86. So, x = 2.

From the second equation, 56 - 3z = 59. So, -3z = 3. Therefore, z = -1.

x = 2, y = -7, z = -1

Answer 2

Solving by repeated substitution:

From second equation:

x=(-5-9z)/2

Substituting for x into 3rd equation:

(-5-9z)/2+4y+6z = -32

scale both sides by 2

-5-9z+8y+12z=-64

y=(-59-3z)/8

Substituting for x and y into the first equation

4(-5-9z)/2 + (59+3z)/8 +3z = 12

-10-18z +(59+3z)/8 + 3z = 12

Scaling both size by 8

-80 -144z + 59+3z + 24z = 96

-117z = 117

z=-1

x=(-5+9)/2 = 4/2 = 2

y = (-59+3)/8 =-56/8 = -7