Verified answer

4x = 5y + 10

z = y - x

3x + 4y = 20 - z

ANSWER:

(5, 2, -3)

Solve the following system:

{4 x = 5 y+10 | (equation 1)

z = y-x | (equation 2)

3 x+4 y = 20-z | (equation 3)

Express the system in standard form:

{4 x-5 y+0 z = 10 | (equation 1)

x-y+z = 0 | (equation 2)

3 x+4 y+z = 20 | (equation 3)

Subtract 1/4 × (equation 1) from equation 2:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+y/4+z = (-5)/2 | (equation 2)

3 x+4 y+z = 20 | (equation 3)

Multiply equation 2 by 4:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+y+4 z = -10 | (equation 2)

3 x+4 y+z = 20 | (equation 3)

Subtract 3/4 × (equation 1) from equation 3:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+y+4 z = -10 | (equation 2)

0 x+(31 y)/4+z = 25/2 | (equation 3)

Multiply equation 3 by 4:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+y+4 z = -10 | (equation 2)

0 x+31 y+4 z = 50 | (equation 3)

Swap equation 2 with equation 3:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+31 y+4 z = 50 | (equation 2)

0 x+y+4 z = -10 | (equation 3)

Subtract 1/31 × (equation 2) from equation 3:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+31 y+4 z = 50 | (equation 2)

0 x+0 y+(120 z)/31 = (-360)/31 | (equation 3)

Multiply equation 3 by 31/120:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+31 y+4 z = 50 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Subtract 4 × (equation 3) from equation 2:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+31 y+0 z = 62 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Divide equation 2 by 31:

{4 x-5 y+0 z = 10 | (equation 1)

0 x+y+0 z = 2 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Add 5 × (equation 2) to equation 1:

{4 x+0 y+0 z = 20 | (equation 1)

0 x+y+0 z = 2 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Divide equation 1 by 4:

{x+0 y+0 z = 5 | (equation 1)

0 x+y+0 z = 2 | (equation 2)

0 x+0 y+z = -3 | (equation 3)

Collect results:

Answer: |

| {x = 5

y = 2

z = -3