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

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

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

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

$x + z = - 3 - 2 y$

$x - 2 y + 2 z = 4$

$2 x - y + z = 5$

Subtract $z$ from both sides of the equation.

$x = - 3 - 2 y - z$

$x - 2 y + 2 z = 4$

$2 x - y + z = 5$

$x = - 3 - 2 y - z$

$x - 2 y + 2 z = 4$

$2 x - y + z = 5$

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

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

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

$x = - 3 - 2 y - z$

$2 x - y + z = 5$

Simplify the left side.

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

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

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

$x = - 3 - 2 y - z$

$2 x - y + z = 5$

Add $- z$ and $2 z$ .

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$2 x - y + z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$2 x - y + z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$2 x - y + z = 5$

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

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Simplify.

Multiply $2$ by $- 3$ .

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Multiply $- 2$ by $2$ .

$- 6 - 4 y + 2 (- z) - y + z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Multiply $- 1$ by $2$ .

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Simplify by adding terms.

Subtract $y$ from $- 4 y$ .

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

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

Add $- 2 z$ and $z$ .

$- 6 - 5 y - z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$- 6 - 5 y - z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$- 6 - 5 y - z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$- 6 - 5 y - z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

$- 6 - 5 y - z = 5$

$- 3 - 4 y + z = 4$

$x = - 3 - 2 y - z$

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

Add $3$ to both sides of the equation.

$- 4 y + z = 4 + 3$

$- 6 - 5 y - z = 5$

$x = - 3 - 2 y - z$

Add $4 y$ to both sides of the equation.

$z = 4 + 3 + 4 y$

$- 6 - 5 y - z = 5$

$x = - 3 - 2 y - z$

Add $4$ and $3$ .

$z = 7 + 4 y$

$- 6 - 5 y - z = 5$

$x = - 3 - 2 y - z$

$z = 7 + 4 y$

$- 6 - 5 y - z = 5$

$x = - 3 - 2 y - z$

Replace all occurrences of $z$ with $7 + 4 y$ in each equation.

Replace all occurrences of $z$ in $- 6 - 5 y - z = 5$ with $7 + 4 y$ .

$- 6 - 5 y - (7 + 4 y) = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Simplify the left side.

Simplify $- 6 - 5 y - (7 + 4 y)$ .

Simplify each term.

Apply the distributive property.

$- 6 - 5 y - 1 \cdot 7 - (4 y) = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Multiply $- 1$ by $7$ .

$- 6 - 5 y - 7 - (4 y) = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Multiply $4$ by $- 1$ .

$- 6 - 5 y - 7 - 4 y = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

$- 6 - 5 y - 7 - 4 y = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Simplify by adding terms.

Subtract $7$ from $- 6$ .

$- 5 y - 13 - 4 y = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Subtract $4 y$ from $- 5 y$ .

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - z$

Replace all occurrences of $z$ in $x = - 3 - 2 y - z$ with $7 + 4 y$ .

$x = - 3 - 2 y - (7 + 4 y)$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Simplify the right side.

Simplify $- 3 - 2 y - (7 + 4 y)$ .

Simplify each term.

Apply the distributive property.

$x = - 3 - 2 y - 1 \cdot 7 - (4 y)$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Multiply $- 1$ by $7$ .

$x = - 3 - 2 y - 7 - (4 y)$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Multiply $4$ by $- 1$ .

$x = - 3 - 2 y - 7 - 4 y$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 3 - 2 y - 7 - 4 y$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Simplify by adding terms.

Subtract $7$ from $- 3$ .

$x = - 2 y - 10 - 4 y$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Subtract $4 y$ from $- 2 y$ .

$x = - 6 y - 10$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 6 y - 10$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 6 y - 10$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 6 y - 10$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

$x = - 6 y - 10$

$- 9 y - 13 = 5$

$z = 7 + 4 y$

Solve for $y$ in $- 9 y - 13 = 5$ .

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

Add $13$ to both sides of the equation.

$- 9 y = 5 + 13$

$x = - 6 y - 10$

$z = 7 + 4 y$

Add $5$ and $13$ .

$- 9 y = 18$

$x = - 6 y - 10$

$z = 7 + 4 y$

$- 9 y = 18$

$x = - 6 y - 10$

$z = 7 + 4 y$

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

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

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

$x = - 6 y - 10$

$z = 7 + 4 y$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

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

$x = - 6 y - 10$

$z = 7 + 4 y$

Divide $y$ by $1$ .

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

$x = - 6 y - 10$

$z = 7 + 4 y$

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

$x = - 6 y - 10$

$z = 7 + 4 y$

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

$x = - 6 y - 10$

$z = 7 + 4 y$

Simplify the right side.

Divide $18$ by $- 9$ .

$y = - 2$

$x = - 6 y - 10$

$z = 7 + 4 y$

$y = - 2$

$x = - 6 y - 10$

$z = 7 + 4 y$

$y = - 2$

$x = - 6 y - 10$

$z = 7 + 4 y$

$y = - 2$

$x = - 6 y - 10$

$z = 7 + 4 y$

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

Replace all occurrences of $y$ in $x = - 6 y - 10$ with $- 2$ .

$x = - 6 \cdot - 2 - 10$

$y = - 2$

$z = 7 + 4 y$

Simplify the right side.

Simplify $- 6 \cdot - 2 - 10$ .

Multiply $- 6$ by $- 2$ .

$x = 12 - 10$

$y = - 2$

$z = 7 + 4 y$

Subtract $10$ from $12$ .

$x = 2$

$y = - 2$

$z = 7 + 4 y$

$x = 2$

$y = - 2$

$z = 7 + 4 y$

$x = 2$

$y = - 2$

$z = 7 + 4 y$

Replace all occurrences of $y$ in $z = 7 + 4 y$ with $- 2$ .

$z = 7 + 4 (- 2)$

$x = 2$

$y = - 2$

Simplify the right side.

Simplify $7 + 4 (- 2)$ .

Multiply $4$ by $- 2$ .

$z = 7 - 8$

$x = 2$

$y = - 2$

Subtract $8$ from $7$ .

$z = - 1$

$x = 2$

$y = - 2$

$z = - 1$

$x = 2$

$y = - 2$

$z = - 1$

$x = 2$

$y = - 2$

$z = - 1$

$x = 2$

$y = - 2$

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

$(2, - 2, - 1)$

The result can be shown in multiple forms.

Point Form:

$(2, - 2, - 1)$

Equation Form:

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

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

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