Problem to solve
Solve \[-4x+y-5z=2\], \[4x+2y+z=1\], \[2x-y+3z=4\]
Solution
STEP 1
Solve the following system:
{-4 x + y - 5 z = 2
4 x + 2 y + z = 1
2 x - y + 3 z = 4
STEP 2
Hint: | Choose an equation and a variable to solve for.
In the first equation, look to solve for y:
{-4 x + y - 5 z = 2
4 x + 2 y + z = 1
2 x - y + 3 z = 4
STEP 3
Hint: | Solve for y.
Add 5 z + 4 x to both sides:
{y = 5 z + 4 x + 2
4 x + 2 y + z = 1
2 x - y + 3 z = 4
STEP 4
Hint: | Perform a substitution.
Substitute y = 5 z + 4 x + 2 into the second and third equations:
{y = 5 z + 4 x + 2
4 x + z + 2 (5 z + 4 x + 2) = 1
-2 z - 2 x - 2 = 4
STEP 5
Hint: | Expand the left hand side of the equation 4 x + z + 2 (5 z + 4 x + 2) = 1.
4 x + z + 2 (5 z + 4 x + 2) = 4 x + z + (10 z + 8 x + 4) = 11 z + 12 x + 4:
{y = 5 z + 4 x + 2
(11 z + 12 x + 4) = 1
-2 z - 2 x - 2 = 4
STEP 6
Hint: | Choose an equation and a variable to solve for.
In the second equation, look to solve for x:
{y = 5 z + 4 x + 2
11 z + 12 x + 4 = 1
-2 z - 2 x - 2 = 4
STEP 7
Hint: | Isolate terms with x to the left hand side.
Subtract 11 z + 4 from both sides:
{y = 5 z + 4 x + 2
12 x = -11 z - 3
-2 z - 2 x - 2 = 4
STEP 8
Hint: | Solve for x.
Divide both sides by 12:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
-2 z - 2 x - 2 = 4
STEP 9
Hint: | Perform a substitution.
Substitute x = -(11 z)/12 - 1/4 into the third equation:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
-2 - 2 (-(11 z)/12 - 1/4) - 2 z = 4
STEP 10
Hint: | Expand the left hand side of the equation -2 - 2 (-(11 z)/12 - 1/4) - 2 z = 4.
-2 - 2 (-(11 z)/12 - 1/4) - 2 z = ((11 z)/6 + 1/2) - 2 z - 2 = -z/6 - 3/2:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
(-z/6 - 3/2) = 4
STEP 11
Hint: | Choose an equation and a variable to solve for.
In the third equation, look to solve for z:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
-z/6 - 3/2 = 4
STEP 12
Hint: | Isolate terms with z to the left hand side.
Add 3/2 to both sides:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
-z/6 = 11/2
STEP 13
Hint: | Solve for z.
Multiply both sides by -6:
{y = 5 z + 4 x + 2
x = -(11 z)/12 - 1/4
z = -33
STEP 14
Hint: | Perform a back substitution.
Substitute z = -33 into the first and second equations:
{y = 4 x - 163
x = 30
z = -33
STEP 15
Hint: | Perform a back substitution.
Substitute x = 30 into the first equation:
{y = -43
x = 30
z = -33
STEP 16
Hint: | Sort results.
Collect results in alphabetical order:
Answer: | 
 | {x = 30
y = -43
z = -33