Solve the System of Equations x+y+2z=-3 2x+y-2z=4 x+2y-z=5

$x + y + 2 z = - 3$ $2 x + y - 2 z = 4$ $x + 2 y - z = 5$

Step 1

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

Subtract $y$ from both sides of the equation.

$x + 2 z = - 3 - y$

$2 x + y - 2 z = 4$

$x + 2 y - z = 5$

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

$x = - 3 - y - 2 z$

$2 x + y - 2 z = 4$

$x + 2 y - z = 5$

$x = - 3 - y - 2 z$

$2 x + y - 2 z = 4$

$x + 2 y - z = 5$

Step 2

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

Replace all occurrences of $x$ in $2 x + y - 2 z = 4$ with $- 3 - y - 2 z$ .

$2 (- 3 - y - 2 z) + y - 2 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 \cdot - 3 + 2 (- y) + 2 (- 2 z) + y - 2 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Simplify.

Multiply $2$ by $- 3$ .

$- 6 + 2 (- y) + 2 (- 2 z) + y - 2 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Multiply $- 1$ by $2$ .

$- 6 - 2 y + 2 (- 2 z) + y - 2 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Multiply $- 2$ by $2$ .

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

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

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

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

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

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Simplify by adding terms.

Add $- 2 y$ and $y$ .

$- 6 - y - 4 z - 2 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

Subtract $2 z$ from $- 4 z$ .

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$x + 2 y - z = 5$

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

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

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Simplify the left side.

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

Add $- y$ and $2 y$ .

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

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Subtract $z$ from $- 2 z$ .

$- 3 + y - 3 z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$- 3 + y - 3 z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$- 3 + y - 3 z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$- 3 + y - 3 z = 5$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Step 3

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

Add $3$ to both sides of the equation.

$y - 3 z = 5 + 3$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Add $3 z$ to both sides of the equation.

$y = 5 + 3 + 3 z$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Add $5$ and $3$ .

$y = 8 + 3 z$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

$y = 8 + 3 z$

$- 6 - y - 6 z = 4$

$x = - 3 - y - 2 z$

Step 4

Replace all occurrences of $y$ with $8 + 3 z$ in each equation.

Replace all occurrences of $y$ in $- 6 - y - 6 z = 4$ with $8 + 3 z$ .

$- 6 - (8 + 3 z) - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Simplify the left side.

Simplify $- 6 - (8 + 3 z) - 6 z$ .

Simplify each term.

Apply the distributive property.

$- 6 - 1 \cdot 8 - (3 z) - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Multiply $- 1$ by $8$ .

$- 6 - 8 - (3 z) - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Multiply $3$ by $- 1$ .

$- 6 - 8 - 3 z - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

$- 6 - 8 - 3 z - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Simplify by adding terms.

Subtract $8$ from $- 6$ .

$- 14 - 3 z - 6 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Subtract $6 z$ from $- 3 z$ .

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 3 - y - 2 z$

Replace all occurrences of $y$ in $x = - 3 - y - 2 z$ with $8 + 3 z$ .

$x = - 3 - (8 + 3 z) - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Simplify the right side.

Simplify $- 3 - (8 + 3 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$x = - 3 - 1 \cdot 8 - (3 z) - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Multiply $- 1$ by $8$ .

$x = - 3 - 8 - (3 z) - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Multiply $3$ by $- 1$ .

$x = - 3 - 8 - 3 z - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 3 - 8 - 3 z - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Simplify by adding terms.

Subtract $8$ from $- 3$ .

$x = - 11 - 3 z - 2 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Subtract $2 z$ from $- 3 z$ .

$x = - 11 - 5 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 11 - 5 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 11 - 5 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 11 - 5 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

$x = - 11 - 5 z$

$- 14 - 9 z = 4$

$y = 8 + 3 z$

Step 5

Solve for $z$ in $- 14 - 9 z = 4$ .

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

Add $14$ to both sides of the equation.

$- 9 z = 4 + 14$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Add $4$ and $14$ .

$- 9 z = 18$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$- 9 z = 18$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Divide each term in $- 9 z = 18$ by $- 9$ and simplify.

Divide each term in $- 9 z = 18$ by $- 9$ .

$\frac{- 9 z}{- 9} = \frac{18}{- 9}$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 z}{- 9} = \frac{18}{- 9}$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Divide $z$ by $1$ .

$z = \frac{18}{- 9}$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$z = \frac{18}{- 9}$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$z = \frac{18}{- 9}$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Simplify the right side.

Divide $18$ by $- 9$ .

$z = - 2$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$z = - 2$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$z = - 2$

$x = - 11 - 5 z$

$y = 8 + 3 z$

$z = - 2$

$x = - 11 - 5 z$

$y = 8 + 3 z$

Step 6

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

Replace all occurrences of $z$ in $x = - 11 - 5 z$ with $- 2$ .

$x = - 11 - 5 \cdot - 2$

$z = - 2$

$y = 8 + 3 z$

Simplify the right side.

Simplify $- 11 - 5 \cdot - 2$ .

Multiply $- 5$ by $- 2$ .

$x = - 11 + 10$

$z = - 2$

$y = 8 + 3 z$

Add $- 11$ and $10$ .

$x = - 1$

$z = - 2$

$y = 8 + 3 z$

$x = - 1$

$z = - 2$

$y = 8 + 3 z$

$x = - 1$

$z = - 2$

$y = 8 + 3 z$

Replace all occurrences of $z$ in $y = 8 + 3 z$ with $- 2$ .

$y = 8 + 3 (- 2)$

$x = - 1$

$z = - 2$

Simplify the right side.

Simplify $8 + 3 (- 2)$ .

Multiply $3$ by $- 2$ .

$y = 8 - 6$

$x = - 1$

$z = - 2$

Subtract $6$ from $8$ .

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

$y = 2$

$x = - 1$

$z = - 2$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 1, 2, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, 2, - 2)$

Equation Form:

$x = - 1, y = 2, z = - 2$

Solve the System of Equations x+y+2z=-3 2x+y-2z=4 x+2y-z=5

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