Solve the System of Equations x+y-2z=5 -x+2y+z=2 2x+3y-z=9

$x + y - 2 z = 5$ $- x + 2 y + z = 2$ $2 x + 3 y - z = 9$

Step 1

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

Subtract $y$ from both sides of the equation.

$x - 2 z = 5 - y$

$- x + 2 y + z = 2$

$2 x + 3 y - z = 9$

Add $2 z$ to both sides of the equation.

$x = 5 - y + 2 z$

$- x + 2 y + z = 2$

$2 x + 3 y - z = 9$

$x = 5 - y + 2 z$

$- x + 2 y + z = 2$

$2 x + 3 y - z = 9$

Step 2

Replace all occurrences of $x$ with $5 - y + 2 z$ in each equation.

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

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 1 \cdot 5 + y - (2 z) + 2 y + z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Simplify.

Multiply $- 1$ by $5$ .

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$- 5 + 1 y - (2 z) + 2 y + z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Multiply $y$ by $1$ .

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Multiply $2$ by $- 1$ .

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Simplify by adding terms.

Add $y$ and $2 y$ .

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

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Add $- 2 z$ and $z$ .

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$2 x + 3 y - z = 9$

Replace all occurrences of $x$ in $2 x + 3 y - z = 9$ with $5 - y + 2 z$ .

$2 (5 - y + 2 z) + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 \cdot 5 + 2 (- y) + 2 (2 z) + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Simplify.

Multiply $2$ by $5$ .

$10 + 2 (- y) + 2 (2 z) + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Multiply $- 1$ by $2$ .

$10 - 2 y + 2 (2 z) + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Multiply $2$ by $2$ .

$10 - 2 y + 4 z + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 - 2 y + 4 z + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 - 2 y + 4 z + 3 y - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Simplify by adding terms.

Add $- 2 y$ and $3 y$ .

$10 + y + 4 z - z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Subtract $z$ from $4 z$ .

$10 + y + 3 z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 + y + 3 z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 + y + 3 z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 + y + 3 z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$10 + y + 3 z = 9$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Step 3

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

Subtract $10$ from both sides of the equation.

$y + 3 z = 9 - 10$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

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

$y = 9 - 10 - 3 z$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Subtract $10$ from $9$ .

$y = - 1 - 3 z$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

$y = - 1 - 3 z$

$- 5 + 3 y - z = 2$

$x = 5 - y + 2 z$

Step 4

Replace all occurrences of $y$ with $- 1 - 3 z$ in each equation.

Replace all occurrences of $y$ in $- 5 + 3 y - z = 2$ with $- 1 - 3 z$ .

$- 5 + 3 (- 1 - 3 z) - z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Simplify the left side.

Simplify $- 5 + 3 (- 1 - 3 z) - z$ .

Simplify each term.

Apply the distributive property.

$- 5 + 3 \cdot - 1 + 3 (- 3 z) - z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Multiply $3$ by $- 1$ .

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

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Multiply $- 3$ by $3$ .

$- 5 - 3 - 9 z - z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

$- 5 - 3 - 9 z - z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Simplify by adding terms.

Subtract $3$ from $- 5$ .

$- 8 - 9 z - z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Subtract $z$ from $- 9 z$ .

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 5 - y + 2 z$

Replace all occurrences of $y$ in $x = 5 - y + 2 z$ with $- 1 - 3 z$ .

$x = 5 - (- 1 - 3 z) + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Simplify the right side.

Simplify $5 - (- 1 - 3 z) + 2 z$ .

Simplify each term.

Apply the distributive property.

$x = 5 + 1 - (- 3 z) + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Multiply $- 1$ by $- 1$ .

$x = 5 + 1 - (- 3 z) + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Multiply $- 3$ by $- 1$ .

$x = 5 + 1 + 3 z + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 5 + 1 + 3 z + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Simplify by adding terms.

Add $5$ and $1$ .

$x = 6 + 3 z + 2 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Add $3 z$ and $2 z$ .

$x = 6 + 5 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 6 + 5 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 6 + 5 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 6 + 5 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

$x = 6 + 5 z$

$- 8 - 10 z = 2$

$y = - 1 - 3 z$

Step 5

Solve for $z$ in $- 8 - 10 z = 2$ .

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

Add $8$ to both sides of the equation.

$- 10 z = 2 + 8$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Add $2$ and $8$ .

$- 10 z = 10$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$- 10 z = 10$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Divide each term in $- 10 z = 10$ by $- 10$ and simplify.

Divide each term in $- 10 z = 10$ by $- 10$ .

$\frac{- 10 z}{- 10} = \frac{10}{- 10}$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 z}{- 10} = \frac{10}{- 10}$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Divide $z$ by $1$ .

$z = \frac{10}{- 10}$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$z = \frac{10}{- 10}$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$z = \frac{10}{- 10}$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Simplify the right side.

Divide $10$ by $- 10$ .

$z = - 1$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$z = - 1$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$z = - 1$

$x = 6 + 5 z$

$y = - 1 - 3 z$

$z = - 1$

$x = 6 + 5 z$

$y = - 1 - 3 z$

Step 6

Replace all occurrences of $z$ with $- 1$ in each equation.

Replace all occurrences of $z$ in $x = 6 + 5 z$ with $- 1$ .

$x = 6 + 5 (- 1)$

$z = - 1$

$y = - 1 - 3 z$

Simplify the right side.

Simplify $6 + 5 (- 1)$ .

Multiply $5$ by $- 1$ .

$x = 6 - 5$

$z = - 1$

$y = - 1 - 3 z$

Subtract $5$ from $6$ .

$x = 1$

$z = - 1$

$y = - 1 - 3 z$

$x = 1$

$z = - 1$

$y = - 1 - 3 z$

$x = 1$

$z = - 1$

$y = - 1 - 3 z$

Replace all occurrences of $z$ in $y = - 1 - 3 z$ with $- 1$ .

$y = - 1 - 3 \cdot - 1$

$x = 1$

$z = - 1$

Simplify the right side.

Simplify $- 1 - 3 \cdot - 1$ .

Multiply $- 3$ by $- 1$ .

$y = - 1 + 3$

$x = 1$

$z = - 1$

Add $- 1$ and $3$ .

$y = 2$

$x = 1$

$z = - 1$

$y = 2$

$x = 1$

$z = - 1$

$y = 2$

$x = 1$

$z = - 1$

$y = 2$

$x = 1$

$z = - 1$

Step 7

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

$(1, 2, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, 2, - 1)$

Equation Form:

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

Solve the System of Equations x+y-2z=5 -x+2y+z=2 2x+3y-z=9

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