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

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

Step 1

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

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

$y + z = - 3 - 2 x$

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

$- x + y + 2 z = 5$

Subtract $z$ from both sides of the equation.

$y = - 3 - 2 x - z$

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

$- x + y + 2 z = 5$

$y = - 3 - 2 x - z$

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

$- x + y + 2 z = 5$

Step 2

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

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

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Simplify.

Multiply $2$ by $- 3$ .

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Multiply $- 2$ by $2$ .

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Multiply $- 1$ by $2$ .

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Simplify by adding terms.

Subtract $4 x$ from $- 2 x$ .

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

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

Add $- 2 z$ and $z$ .

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- x + y + 2 z = 5$

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

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

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Simplify the left side.

Simplify $- x - 3 - 2 x - z + 2 z$ .

Remove parentheses.

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

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Subtract $2 x$ from $- x$ .

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

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Add $- z$ and $2 z$ .

$- 3 x - 3 + z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- 3 x - 3 + z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- 3 x - 3 + z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$- 3 x - 3 + z = 5$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Step 3

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

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

$- 3 + z = 5 + 3 x$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Add $3$ to both sides of the equation.

$z = 5 + 3 x + 3$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Add $5$ and $3$ .

$z = 3 x + 8$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

$z = 3 x + 8$

$- 6 x - 6 - z = 4$

$y = - 3 - 2 x - z$

Step 4

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

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

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

$z = 3 x + 8$

$y = - 3 - 2 x - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$z = 3 x + 8$

$y = - 3 - 2 x - z$

Multiply $3$ by $- 1$ .

$- 6 x - 6 - 3 x - 1 \cdot 8 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

Multiply $- 1$ by $8$ .

$- 6 x - 6 - 3 x - 8 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

$- 6 x - 6 - 3 x - 8 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

Simplify by adding terms.

Subtract $3 x$ from $- 6 x$ .

$- 9 x - 6 - 8 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

Subtract $8$ from $- 6$ .

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - z$

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

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

$- 9 x - 14 = 4$

$z = 3 x + 8$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$- 9 x - 14 = 4$

$z = 3 x + 8$

Multiply $3$ by $- 1$ .

$y = - 3 - 2 x - 3 x - 1 \cdot 8$

$- 9 x - 14 = 4$

$z = 3 x + 8$

Multiply $- 1$ by $8$ .

$y = - 3 - 2 x - 3 x - 8$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 3 - 2 x - 3 x - 8$

$- 9 x - 14 = 4$

$z = 3 x + 8$

Simplify by adding terms.

Subtract $8$ from $- 3$ .

$y = - 2 x - 3 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

Subtract $3 x$ from $- 2 x$ .

$y = - 5 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 5 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 5 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 5 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

$y = - 5 x - 11$

$- 9 x - 14 = 4$

$z = 3 x + 8$

Step 5

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

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

Add $14$ to both sides of the equation.

$- 9 x = 4 + 14$

$y = - 5 x - 11$

$z = 3 x + 8$

Add $4$ and $14$ .

$- 9 x = 18$

$y = - 5 x - 11$

$z = 3 x + 8$

$- 9 x = 18$

$y = - 5 x - 11$

$z = 3 x + 8$

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

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

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

$y = - 5 x - 11$

$z = 3 x + 8$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

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

$y = - 5 x - 11$

$z = 3 x + 8$

Divide $x$ by $1$ .

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

$y = - 5 x - 11$

$z = 3 x + 8$

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

$y = - 5 x - 11$

$z = 3 x + 8$

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

$y = - 5 x - 11$

$z = 3 x + 8$

Simplify the right side.

Divide $18$ by $- 9$ .

$x = - 2$

$y = - 5 x - 11$

$z = 3 x + 8$

$x = - 2$

$y = - 5 x - 11$

$z = 3 x + 8$

$x = - 2$

$y = - 5 x - 11$

$z = 3 x + 8$

$x = - 2$

$y = - 5 x - 11$

$z = 3 x + 8$

Step 6

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

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

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

$x = - 2$

$z = 3 x + 8$

Simplify the right side.

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

Multiply $- 5$ by $- 2$ .

$y = 10 - 11$

$x = - 2$

$z = 3 x + 8$

Subtract $11$ from $10$ .

$y = - 1$

$x = - 2$

$z = 3 x + 8$

$y = - 1$

$x = - 2$

$z = 3 x + 8$

$y = - 1$

$x = - 2$

$z = 3 x + 8$

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

$z = 3 (- 2) + 8$

$y = - 1$

$x = - 2$

Simplify the right side.

Simplify $3 (- 2) + 8$ .

Multiply $3$ by $- 2$ .

$z = - 6 + 8$

$y = - 1$

$x = - 2$

Add $- 6$ and $8$ .

$z = 2$

$y = - 1$

$x = - 2$

$z = 2$

$y = - 1$

$x = - 2$

$z = 2$

$y = - 1$

$x = - 2$

$z = 2$

$y = - 1$

$x = - 2$

Step 7

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

$(- 2, - 1, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 2, - 1, 2)$

Equation Form:

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

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

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