Answer:
x = 1
, y = 1
, z = -3
Step-by-step explanation:
Solve the following system:
{3 x + 2 y - z = 8
2 x + 2 z = -4
x + 3 y = 4
Hint: | Choose an equation and a variable to solve for.
In the third equation, look to solve for x:
{3 x + 2 y - z = 8
2 x + 2 z = -4
x + 3 y = 4
Hint: | Solve for x.
Subtract 3 y from both sides:
{3 x + 2 y - z = 8
2 x + 2 z = -4
x = 4 - 3 y
Hint: | Perform a substitution.
Substitute x = 4 - 3 y into the first and second equations:
{3 (4 - 3 y) + 2 y - z = 8
2 (4 - 3 y) + 2 z = -4
x = 4 - 3 y
Hint: | Expand the left hand side of the equation 3 (4 - 3 y) + 2 y - z = 8.
3 (4 - 3 y) + 2 y - z = (12 - 9 y) + 2 y - z = 12 - 7 y - z:
{12 - 7 y - z = 8
2 (4 - 3 y) + 2 z = -4
x = 4 - 3 y
Hint: | Expand the left hand side of the equation 2 (4 - 3 y) + 2 z = -4.
2 (4 - 3 y) + 2 z = (8 - 6 y) + 2 z = 8 - 6 y + 2 z:
{12 - 7 y - z = 8
8 - 6 y + 2 z = -4
x = 4 - 3 y
Hint: | Choose an equation and a variable to solve for.
In the first equation, look to solve for y:
{12 - 7 y - z = 8
8 - 6 y + 2 z = -4
x = 4 - 3 y
Hint: | Isolate terms with y to the left hand side.
Subtract 12 - z from both sides:
{-7 y = z - 4
8 - 6 y + 2 z = -4
x = 4 - 3 y
Hint: | Solve for y.
Divide both sides by -7:
{y = 4/7 - z/7
8 - 6 y + 2 z = -4
x = 4 - 3 y
Hint: | Perform a substitution.
Substitute y = 4/7 - z/7 into the second equation:
{y = 4/7 - z/7
8 - 6 (4/7 - z/7) + 2 z = -4
x = 4 - 3 y
Hint: | Expand the left hand side of the equation 8 - 6 (4/7 - z/7) + 2 z = -4.
8 - 6 (4/7 - z/7) + 2 z = 2 z + ((6 z)/7 - 24/7) + 8 = (20 z)/7 + 32/7:
{y = 4/7 - z/7
(20 z)/7 + 32/7 = -4
x = 4 - 3 y
Hint: | Choose an equation and a variable to solve for.
In the second equation, look to solve for z:
{y = 4/7 - z/7
(20 z)/7 + 32/7 = -4
x = 4 - 3 y
Hint: | Isolate terms with z to the left hand side.
Subtract 32/7 from both sides:
{y = 4/7 - z/7
(20 z)/7 = -60/7
x = 4 - 3 y
Hint: | Solve for z.
Multiply both sides by 7/20:
{y = 4/7 - z/7
z = -3
x = 4 - 3 y
Hint: | Perform a back substitution.
Substitute z = -3 into the first equation:
{y = 1
z = -3
x = 4 - 3 y
Hint: | Perform a back substitution.
Substitute y = 1 into the third equation:
{y = 1
z = -3
x = 1
Hint: | Sort results.
Collect results in alphabetical order:
Answer: {x = 1
, y = 1
, z = -3
