Answer:
x = 3
, y = 6
, z = -1
Step-by-step explanation using substitution:
Solve the following system:
{-x - y - z = -8
-4 x + 4 y + 5 z = 7
2 x + 2 z = 4
In the first equation, look to solve for x:
{-x - y - z = -8
-4 x + 4 y + 5 z = 7
2 x + 2 z = 4
Add y + z to both sides:
{-x = -8 + y + z
-4 x + 4 y + 5 z = 7
2 x + 2 z = 4
Multiply both sides by -1:
{x = 8 - y - z
-4 x + 4 y + 5 z = 7
2 x + 2 z = 4
Substitute x = 8 - y - z into the second and third equations:
{x = 8 - y - z
4 y - 4 (8 - y - z) + 5 z = 7
2 (8 - y - z) + 2 z = 4
4 y - 4 (8 - y - z) + 5 z = 4 y + (-32 + 4 y + 4 z) + 5 z = -32 + 8 y + 9 z:
{x = 8 - y - z
-32 + 8 y + 9 z = 7
2 (8 - y - z) + 2 z = 4
2 (8 - y - z) + 2 z = (16 - 2 y - 2 z) + 2 z = 16 - 2 y:
{x = 8 - y - z
-32 + 8 y + 9 z = 7
16 - 2 y = 4
In the third equation, look to solve for y:
{x = 8 - y - z
-32 + 8 y + 9 z = 7
16 - 2 y = 4
Subtract 16 from both sides:
{x = 8 - y - z
-32 + 8 y + 9 z = 7
-2 y = -12
Divide both sides by -2:
{x = 8 - y - z
-32 + 8 y + 9 z = 7
y = 6
Substitute y = 6 into the second equation:
{x = 8 - y - z
9 z + 16 = 7
y = 6
In the second equation, look to solve for z:
{x = 8 - y - z
9 z + 16 = 7
y = 6
Subtract 16 from both sides:
{x = 8 - y - z
9 z = -9
y = 6
Divide both sides by 9:
{x = 8 - y - z
z = -1
y = 6
Substitute z = -1 into the first equation:
{x = 9 - y
z = -1
y = 6
Substitute y = 6 into the first equation:
{x = 3
z = -1
y = 6
Collect results in alphabetical order:
Answer: {x = 3
, y = 6
, z = -1
