Problem to solve
Solve \[-10x+3y+z=-2\], \[3x-2y+5z=4\], \[-6x+4y+2z=1\]
Solution
STEP 1
Solve the following system:
{-10 x + 3 y + z = -2
3 x - 2 y + 5 z = 4
-6 x + 4 y + 2 z = 1
STEP 2
Hint: | Choose an equation and a variable to solve for.
In the first equation, look to solve for z:
{-10 x + 3 y + z = -2
3 x - 2 y + 5 z = 4
-6 x + 4 y + 2 z = 1
STEP 3
Hint: | Solve for z.
Subtract 3 y - 10 x from both sides:
{z = 10 x + (-3 y - 2)
3 x - 2 y + 5 z = 4
-6 x + 4 y + 2 z = 1
STEP 4
Hint: | Perform a substitution.
Substitute z = -3 y + 10 x - 2 into the second and third equations:
{z = -3 y + 10 x - 2
3 x + 5 (-3 y + 10 x - 2) - 2 y = 4
-6 x + 2 (-3 y + 10 x - 2) + 4 y = 1
STEP 5
Hint: | Expand the left hand side of the equation 3 x + 5 (-3 y + 10 x - 2) - 2 y = 4.
3 x + 5 (-3 y + 10 x - 2) - 2 y = 3 x + (-15 y + 50 x - 10) - 2 y = -17 y + 53 x - 10:
{z = -3 y + 10 x - 2
(-17 y + 53 x - 10) = 4
-6 x + 2 (-3 y + 10 x - 2) + 4 y = 1
STEP 6
Hint: | Expand the left hand side of the equation -6 x + 2 (-3 y + 10 x - 2) + 4 y = 1.
-6 x + 2 (-3 y + 10 x - 2) + 4 y = -6 x + (-6 y + 20 x - 4) + 4 y = -2 y + 14 x - 4:
{z = -3 y + 10 x - 2
-17 y + 53 x - 10 = 4
(-2 y + 14 x - 4) = 1
STEP 7
Hint: | Choose an equation and a variable to solve for.
In the second equation, look to solve for x:
{z = -3 y + 10 x - 2
-17 y + 53 x - 10 = 4
-2 y + 14 x - 4 = 1
STEP 8
Hint: | Isolate terms with x to the left hand side.
Add 17 y + 10 to both sides:
{z = -3 y + 10 x - 2
53 x = 17 y + 14
-2 y + 14 x - 4 = 1
STEP 9
Hint: | Solve for x.
Divide both sides by 53:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
-2 y + 14 x - 4 = 1
STEP 10
Hint: | Perform a substitution.
Substitute x = (17 y)/53 + 14/53 into the third equation:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
-4 + 14 ((17 y)/53 + 14/53) - 2 y = 1
STEP 11
Hint: | Expand the left hand side of the equation -4 + 14 ((17 y)/53 + 14/53) - 2 y = 1.
-4 + 14 ((17 y)/53 + 14/53) - 2 y = ((238 y)/53 + 196/53) - 2 y - 4 = (132 y)/53 - 16/53:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
((132 y)/53 - 16/53) = 1
STEP 12
Hint: | Choose an equation and a variable to solve for.
In the third equation, look to solve for y:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
(132 y)/53 - 16/53 = 1
STEP 13
Hint: | Isolate terms with y to the left hand side.
Add 16/53 to both sides:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
(132 y)/53 = 69/53
STEP 14
Hint: | Solve for y.
Multiply both sides by 53/132:
{z = -3 y + 10 x - 2
x = (17 y)/53 + 14/53
y = 23/44
STEP 15
Hint: | Perform a back substitution.
Substitute y = 23/44 into the first and second equations:
{z = 10 x - 157/44
x = 19/44
y = 23/44
STEP 16
Hint: | Perform a back substitution.
Substitute x = 19/44 into the first equation:
{z = 3/4
x = 19/44
y = 23/44
STEP 17
Hint: | Sort results.
Collect results in alphabetical order:
Answer: | 
 | {x = 19/44
y = 23/44
z = 3/4