Solve the System of Equations x-y+z=0 -x+2y-3z=-5 2x-3y+5z=8

$x - y + z = 0$ $- x + 2 y - 3 z = - 5$ $2 x - 3 y + 5 z = 8$

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

Add $y$ to both sides of the equation.

$x + z = y$

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

$2 x - 3 y + 5 z = 8$

Subtract $z$ from both sides of the equation.

$x = y - z$

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

$2 x - 3 y + 5 z = 8$

$x = y - z$

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

$2 x - 3 y + 5 z = 8$

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

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

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

$x = y - z$

$2 x - 3 y + 5 z = 8$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = y - z$

$2 x - 3 y + 5 z = 8$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$- y + 1 z + 2 y - 3 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

Multiply $z$ by $1$ .

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

$x = y - z$

$2 x - 3 y + 5 z = 8$

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

$x = y - z$

$2 x - 3 y + 5 z = 8$

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

$x = y - z$

$2 x - 3 y + 5 z = 8$

Simplify by adding terms.

Add $- y$ and $2 y$ .

$y + z - 3 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

Subtract $3 z$ from $z$ .

$y - 2 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

$2 x - 3 y + 5 z = 8$

Replace all occurrences of $x$ in $2 x - 3 y + 5 z = 8$ with $y - z$ .

$2 (y - z) - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 y + 2 (- z) - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

Multiply $- 1$ by $2$ .

$2 y - 2 z - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

$2 y - 2 z - 3 y + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

Simplify by adding terms.

Subtract $3 y$ from $2 y$ .

$- y - 2 z + 5 z = 8$

$y - 2 z = - 5$

$x = y - z$

Add $- 2 z$ and $5 z$ .

$- y + 3 z = 8$

$y - 2 z = - 5$

$x = y - z$

$- y + 3 z = 8$

$y - 2 z = - 5$

$x = y - z$

$- y + 3 z = 8$

$y - 2 z = - 5$

$x = y - z$

$- y + 3 z = 8$

$y - 2 z = - 5$

$x = y - z$

$- y + 3 z = 8$

$y - 2 z = - 5$

$x = y - z$

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

$y = - 5 + 2 z$

$- y + 3 z = 8$

$x = y - z$

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

Replace all occurrences of $y$ in $- y + 3 z = 8$ with $- 5 + 2 z$ .

$- (- 5 + 2 z) + 3 z = 8$

$y = - 5 + 2 z$

$x = y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$5 - (2 z) + 3 z = 8$

$y = - 5 + 2 z$

$x = y - z$

Multiply $- 1$ by $- 5$ .

$5 - (2 z) + 3 z = 8$

$y = - 5 + 2 z$

$x = y - z$

Multiply $2$ by $- 1$ .

$5 - 2 z + 3 z = 8$

$y = - 5 + 2 z$

$x = y - z$

$5 - 2 z + 3 z = 8$

$y = - 5 + 2 z$

$x = y - z$

Add $- 2 z$ and $3 z$ .

$5 + z = 8$

$y = - 5 + 2 z$

$x = y - z$

$5 + z = 8$

$y = - 5 + 2 z$

$x = y - z$

$5 + z = 8$

$y = - 5 + 2 z$

$x = y - z$

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

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

$5 + z = 8$

$y = - 5 + 2 z$

Simplify the right side.

Subtract $z$ from $2 z$ .

$x = - 5 + z$

$5 + z = 8$

$y = - 5 + 2 z$

$x = - 5 + z$

$5 + z = 8$

$y = - 5 + 2 z$

$x = - 5 + z$

$5 + z = 8$

$y = - 5 + 2 z$

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

Subtract $5$ from both sides of the equation.

$z = 8 - 5$

$x = - 5 + z$

$y = - 5 + 2 z$

Subtract $5$ from $8$ .

$z = 3$

$x = - 5 + z$

$y = - 5 + 2 z$

$z = 3$

$x = - 5 + z$

$y = - 5 + 2 z$

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

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

$x = - 5 + 3$

$z = 3$

$y = - 5 + 2 z$

Simplify $x = - 5 + 3$ .

Simplify the left side.

Remove parentheses.

$x = - 5 + 3$

$z = 3$

$y = - 5 + 2 z$

$x = - 5 + 3$

$z = 3$

$y = - 5 + 2 z$

Simplify the right side.

Add $- 5$ and $3$ .

$x = - 2$

$z = 3$

$y = - 5 + 2 z$

$x = - 2$

$z = 3$

$y = - 5 + 2 z$

$x = - 2$

$z = 3$

$y = - 5 + 2 z$

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

$y = - 5 + 2 (3)$

$x = - 2$

$z = 3$

Simplify the right side.

Simplify $- 5 + 2 (3)$ .

Multiply $2$ by $3$ .

$y = - 5 + 6$

$x = - 2$

$z = 3$

Add $- 5$ and $6$ .

$y = 1$

$x = - 2$

$z = 3$

$y = 1$

$x = - 2$

$z = 3$

$y = 1$

$x = - 2$

$z = 3$

$y = 1$

$x = - 2$

$z = 3$

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

$(- 2, 1, 3)$

The result can be shown in multiple forms.

Point Form:

$(- 2, 1, 3)$

Equation Form:

$x = - 2, y = 1, z = 3$

Solve the System of Equations x-y+z=0 -x+2y-3z=-5 2x-3y+5z=8

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