Solve the System of Equations x+2y-z=-3 -x+y+2z=5 -2x+5y+5z=12

$x + 2 y - z = - 3$ $- x + y + 2 z = 5$ $- 2 x + 5 y + 5 z = 12$

Step 1

Move all terms not containing $x$ to the right side of the equation.

Subtract $2 y$ from both sides of the equation.

$x - z = - 3 - 2 y$

$- x + y + 2 z = 5$

$- 2 x + 5 y + 5 z = 12$

Add $z$ to both sides of the equation.

$x = - 3 - 2 y + z$

$- x + y + 2 z = 5$

$- 2 x + 5 y + 5 z = 12$

$x = - 3 - 2 y + z$

$- x + y + 2 z = 5$

$- 2 x + 5 y + 5 z = 12$

Step 2

Replace all occurrences of $x$ with $- 3 - 2 y + z$ in each equation.

Replace all occurrences of $x$ in $- x + y + 2 z = 5$ with $- 3 - 2 y + z$ .

$- (- 3 - 2 y + z) + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Simplify the left side.

Simplify $- (- 3 - 2 y + z) + y + 2 z$ .

Simplify each term.

Apply the distributive property.

$3 - (- 2 y) - z + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Simplify.

Multiply $- 1$ by $- 3$ .

$3 - (- 2 y) - z + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Multiply $- 2$ by $- 1$ .

$3 + 2 y - z + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

$3 + 2 y - z + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

$3 + 2 y - z + y + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Simplify by adding terms.

Add $2 y$ and $y$ .

$3 + 3 y - z + 2 z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Add $- z$ and $2 z$ .

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$- 2 x + 5 y + 5 z = 12$

Replace all occurrences of $x$ in $- 2 x + 5 y + 5 z = 12$ with $- 3 - 2 y + z$ .

$- 2 (- 3 - 2 y + z) + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Simplify the left side.

Simplify $- 2 (- 3 - 2 y + z) + 5 y + 5 z$ .

Simplify each term.

Apply the distributive property.

$- 2 \cdot - 3 - 2 (- 2 y) - 2 z + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Simplify.

Multiply $- 2$ by $- 3$ .

$6 - 2 (- 2 y) - 2 z + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Multiply $- 2$ by $- 2$ .

$6 + 4 y - 2 z + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 4 y - 2 z + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 4 y - 2 z + 5 y + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Simplify by adding terms.

Add $4 y$ and $5 y$ .

$6 + 9 y - 2 z + 5 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Add $- 2 z$ and $5 z$ .

$6 + 9 y + 3 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 9 y + 3 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 9 y + 3 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 9 y + 3 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

$6 + 9 y + 3 z = 12$

$3 + 3 y + z = 5$

$x = - 3 - 2 y + z$

Step 3

Move all terms not containing $z$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$3 y + z = 5 - 3$

$6 + 9 y + 3 z = 12$

$x = - 3 - 2 y + z$

Subtract $3 y$ from both sides of the equation.

$z = 5 - 3 - 3 y$

$6 + 9 y + 3 z = 12$

$x = - 3 - 2 y + z$

Subtract $3$ from $5$ .

$z = 2 - 3 y$

$6 + 9 y + 3 z = 12$

$x = - 3 - 2 y + z$

$z = 2 - 3 y$

$6 + 9 y + 3 z = 12$

$x = - 3 - 2 y + z$

Step 4

Replace all occurrences of $z$ with $2 - 3 y$ in each equation.

Replace all occurrences of $z$ in $6 + 9 y + 3 z = 12$ with $2 - 3 y$ .

$6 + 9 y + 3 (2 - 3 y) = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Simplify the left side.

Simplify $6 + 9 y + 3 (2 - 3 y)$ .

Simplify each term.

Apply the distributive property.

$6 + 9 y + 3 \cdot 2 + 3 (- 3 y) = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Multiply $3$ by $2$ .

$6 + 9 y + 6 + 3 (- 3 y) = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Multiply $- 3$ by $3$ .

$6 + 9 y + 6 - 9 y = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

$6 + 9 y + 6 - 9 y = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Simplify by adding terms.

Combine the opposite terms in $6 + 9 y + 6 - 9 y$ .

Subtract $9 y$ from $9 y$ .

$6 + 0 + 6 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Add $6$ and $0$ .

$6 + 6 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

$6 + 6 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Add $6$ and $6$ .

$12 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

$12 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

$12 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

$12 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + z$

Replace all occurrences of $z$ in $x = - 3 - 2 y + z$ with $2 - 3 y$ .

$x = - 3 - 2 y + 2 - 3 y$

$12 = 12$

$z = 2 - 3 y$

Simplify $x = - 3 - 2 y + 2 - 3 y$ .

Simplify the left side.

Remove parentheses.

$x = - 3 - 2 y + 2 - 3 y$

$12 = 12$

$z = 2 - 3 y$

$x = - 3 - 2 y + 2 - 3 y$

$12 = 12$

$z = 2 - 3 y$

Simplify the right side.

Simplify $- 3 - 2 y + 2 - 3 y$ .

Add $- 3$ and $2$ .

$x = - 2 y - 1 - 3 y$

$12 = 12$

$z = 2 - 3 y$

Subtract $3 y$ from $- 2 y$ .

$x = - 5 y - 1$

$12 = 12$

$z = 2 - 3 y$

$x = - 5 y - 1$

$12 = 12$

$z = 2 - 3 y$

$x = - 5 y - 1$

$12 = 12$

$z = 2 - 3 y$

$x = - 5 y - 1$

$12 = 12$

$z = 2 - 3 y$

$x = - 5 y - 1$

$12 = 12$

$z = 2 - 3 y$

Step 5

Remove any equations from the system that are always true.

$x = - 5 y - 1$

$z = 2 - 3 y$

Solve the System of Equations x+2y-z=-3 -x+y+2z=5 -2x+5y+5z=12

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