Solve the System of Equations -x+2y+z=0 -x+2y+3z=0 x-5y-2z=9

$- x + 2 y + z = 0$ $- x + 2 y + 3 z = 0$ $x - 5 y - 2 z = 9$

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

Add $x$ to both sides of the equation.

$2 y + z = x$

$- x + 2 y + 3 z = 0$

$x - 5 y - 2 z = 9$

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

$z = x - 2 y$

$- x + 2 y + 3 z = 0$

$x - 5 y - 2 z = 9$

$z = x - 2 y$

$- x + 2 y + 3 z = 0$

$x - 5 y - 2 z = 9$

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

Replace all occurrences of $z$ in $- x + 2 y + 3 z = 0$ with $x - 2 y$ .

$- x + 2 y + 3 (x - 2 y) = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- x + 2 y + 3 x + 3 (- 2 y) = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

Multiply $- 2$ by $3$ .

$- x + 2 y + 3 x - 6 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

$- x + 2 y + 3 x - 6 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

Simplify by adding terms.

Add $- x$ and $3 x$ .

$2 x + 2 y - 6 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

Subtract $6 y$ from $2 y$ .

$2 x - 4 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$x - 5 y - 2 z = 9$

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

$x - 5 y - 2 (x - 2 y) = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

Simplify the left side.

Simplify $x - 5 y - 2 (x - 2 y)$ .

Simplify each term.

Apply the distributive property.

$x - 5 y - 2 x - 2 (- 2 y) = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

Multiply $- 2$ by $- 2$ .

$x - 5 y - 2 x + 4 y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$x - 5 y - 2 x + 4 y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

Simplify by adding terms.

Subtract $2 x$ from $x$ .

$- x - 5 y + 4 y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

Add $- 5 y$ and $4 y$ .

$- x - y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$- x - y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$- x - y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$- x - y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

$- x - y = 9$

$2 x - 4 y = 0$

$z = x - 2 y$

Solve for $x$ in $- x - y = 9$ .

Add $y$ to both sides of the equation.

$- x = 9 + y$

$2 x - 4 y = 0$

$z = x - 2 y$

Divide each term in $- x = 9 + y$ by $- 1$ and simplify.

Divide each term in $- x = 9 + y$ by $- 1$ .

$\frac{- x}{- 1} = \frac{9}{- 1} + \frac{y}{- 1}$

$2 x - 4 y = 0$

$z = x - 2 y$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{9}{- 1} + \frac{y}{- 1}$

$2 x - 4 y = 0$

$z = x - 2 y$

Divide $x$ by $1$ .

$x = \frac{9}{- 1} + \frac{y}{- 1}$

$2 x - 4 y = 0$

$z = x - 2 y$

$x = \frac{9}{- 1} + \frac{y}{- 1}$

$2 x - 4 y = 0$

$z = x - 2 y$

Simplify the right side.

Simplify each term.

Divide $9$ by $- 1$ .

$x = - 9 + \frac{y}{- 1}$

$2 x - 4 y = 0$

$z = x - 2 y$

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$x = - 9 - 1 \cdot y$

$2 x - 4 y = 0$

$z = x - 2 y$

Rewrite $- 1 \cdot y$ as $- y$ .

$x = - 9 - y$

$2 x - 4 y = 0$

$z = x - 2 y$

$x = - 9 - y$

$2 x - 4 y = 0$

$z = x - 2 y$

$x = - 9 - y$

$2 x - 4 y = 0$

$z = x - 2 y$

$x = - 9 - y$

$2 x - 4 y = 0$

$z = x - 2 y$

$x = - 9 - y$

$2 x - 4 y = 0$

$z = x - 2 y$

Replace all occurrences of $x$ with $- 9 - y$ in each equation.

Replace all occurrences of $x$ in $2 x - 4 y = 0$ with $- 9 - y$ .

$2 (- 9 - y) - 4 y = 0$

$x = - 9 - y$

$z = x - 2 y$

Simplify the left side.

Simplify $2 (- 9 - y) - 4 y$ .

Simplify each term.

Apply the distributive property.

$2 \cdot - 9 + 2 (- y) - 4 y = 0$

$x = - 9 - y$

$z = x - 2 y$

Multiply $2$ by $- 9$ .

$- 18 + 2 (- y) - 4 y = 0$

$x = - 9 - y$

$z = x - 2 y$

Multiply $- 1$ by $2$ .

$- 18 - 2 y - 4 y = 0$

$x = - 9 - y$

$z = x - 2 y$

$- 18 - 2 y - 4 y = 0$

$x = - 9 - y$

$z = x - 2 y$

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

$- 18 - 6 y = 0$

$x = - 9 - y$

$z = x - 2 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

$z = x - 2 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

$z = x - 2 y$

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

$z = (- 9 - y) - 2 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

Simplify the right side.

Subtract $2 y$ from $- y$ .

$z = - 9 - 3 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

$z = - 9 - 3 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

$z = - 9 - 3 y$

$- 18 - 6 y = 0$

$x = - 9 - y$

Solve for $y$ in $- 18 - 6 y = 0$ .

Add $18$ to both sides of the equation.

$- 6 y = 18$

$z = - 9 - 3 y$

$x = - 9 - y$

Divide each term in $- 6 y = 18$ by $- 6$ and simplify.

Divide each term in $- 6 y = 18$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{18}{- 6}$

$z = - 9 - 3 y$

$x = - 9 - y$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{18}{- 6}$

$z = - 9 - 3 y$

$x = - 9 - y$

Divide $y$ by $1$ .

$y = \frac{18}{- 6}$

$z = - 9 - 3 y$

$x = - 9 - y$

$y = \frac{18}{- 6}$

$z = - 9 - 3 y$

$x = - 9 - y$

$y = \frac{18}{- 6}$

$z = - 9 - 3 y$

$x = - 9 - y$

Simplify the right side.

Divide $18$ by $- 6$ .

$y = - 3$

$z = - 9 - 3 y$

$x = - 9 - y$

$y = - 3$

$z = - 9 - 3 y$

$x = - 9 - y$

$y = - 3$

$z = - 9 - 3 y$

$x = - 9 - y$

$y = - 3$

$z = - 9 - 3 y$

$x = - 9 - y$

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

Replace all occurrences of $y$ in $z = - 9 - 3 y$ with $- 3$ .

$z = - 9 - 3 \cdot - 3$

$y = - 3$

$x = - 9 - y$

Simplify the right side.

Simplify $- 9 - 3 \cdot - 3$ .

Multiply $- 3$ by $- 3$ .

$z = - 9 + 9$

$y = - 3$

$x = - 9 - y$

Add $- 9$ and $9$ .

$z = 0$

$y = - 3$

$x = - 9 - y$

$z = 0$

$y = - 3$

$x = - 9 - y$

$z = 0$

$y = - 3$

$x = - 9 - y$

Replace all occurrences of $y$ in $x = - 9 - y$ with $- 3$ .

$x = - 9 - (- 3)$

$z = 0$

$y = - 3$

Simplify the right side.

Simplify $- 9 - (- 3)$ .

Multiply $- 1$ by $- 3$ .

$x = - 9 + 3$

$z = 0$

$y = - 3$

Add $- 9$ and $3$ .

$x = - 6$

$z = 0$

$y = - 3$

$x = - 6$

$z = 0$

$y = - 3$

$x = - 6$

$z = 0$

$y = - 3$

$x = - 6$

$z = 0$

$y = - 3$

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

$(- 6, - 3, 0)$

The result can be shown in multiple forms.

Point Form:

$(- 6, - 3, 0)$

Equation Form:

$x = - 6, y = - 3, z = 0$

Solve the System of Equations -x+2y+z=0 -x+2y+3z=0 x-5y-2z=9

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