Solve the System of Equations x-y+z=-1 x+y-2z=7 -2x+4y+2z=-4

$x - y + z = - 1$ $x + y - 2 z = 7$ $- 2 x + 4 y + 2 z = - 4$

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

Add $y$ to both sides of the equation.

$x + z = - 1 + y$

$x + y - 2 z = 7$

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

Subtract $z$ from both sides of the equation.

$x = - 1 + y - z$

$x + y - 2 z = 7$

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

$x = - 1 + y - z$

$x + y - 2 z = 7$

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

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

Replace all occurrences of $x$ in $x + y - 2 z = 7$ with $- 1 + y - z$ .

$(- 1 + y - z) + y - 2 z = 7$

$x = - 1 + y - z$

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

Simplify the left side.

Simplify $(- 1 + y - z) + y - 2 z$ .

Add $y$ and $y$ .

$- 1 + 2 y - z - 2 z = 7$

$x = - 1 + y - z$

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

Subtract $2 z$ from $- z$ .

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

Replace all occurrences of $x$ in $- 2 x + 4 y + 2 z = - 4$ with $- 1 + y - z$ .

$- 2 (- 1 + y - z) + 4 y + 2 z = - 4$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Simplify the left side.

Simplify $- 2 (- 1 + y - z) + 4 y + 2 z$ .

Simplify each term.

Apply the distributive property.

$- 2 \cdot - 1 - 2 y - 2 (- z) + 4 y + 2 z = - 4$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Simplify.

Multiply $- 2$ by $- 1$ .

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Multiply $- 1$ by $- 2$ .

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Simplify by adding terms.

Add $- 2 y$ and $4 y$ .

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Add $2 z$ and $2 z$ .

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Solve for $y$ in $2 + 2 y + 4 z = - 4$ .

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

Subtract $2$ from both sides of the equation.

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

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Subtract $4 z$ from both sides of the equation.

$2 y = - 4 - 2 - 4 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Subtract $2$ from $- 4$ .

$2 y = - 6 - 4 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$2 y = - 6 - 4 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Divide each term in $2 y = - 6 - 4 z$ by $2$ and simplify.

Divide each term in $2 y = - 6 - 4 z$ by $2$ .

$\frac{2 y}{2} = \frac{- 6}{2} + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 6}{2} + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Divide $y$ by $1$ .

$y = \frac{- 6}{2} + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = \frac{- 6}{2} + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = \frac{- 6}{2} + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Simplify the right side.

Simplify each term.

Divide $- 6$ by $2$ .

$y = - 3 + \frac{- 4 z}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4 z$ .

$y = - 3 + \frac{2 (- 2 z)}{2}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Cancel the common factors.

Factor $2$ out of $2$ .

$y = - 3 + \frac{2 (- 2 z)}{2 (1)}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Cancel the common factor.

$y = - 3 + \frac{2 (- 2 z)}{2 \cdot 1}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Rewrite the expression.

$y = - 3 + \frac{- 2 z}{1}$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

Divide $- 2 z$ by $1$ .

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

$y = - 3 - 2 z$

$- 1 + 2 y - 3 z = 7$

$x = - 1 + y - z$

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

Replace all occurrences of $y$ in $- 1 + 2 y - 3 z = 7$ with $- 3 - 2 z$ .

$- 1 + 2 (- 3 - 2 z) - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 1 + 2 \cdot - 3 + 2 (- 2 z) - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

Multiply $2$ by $- 3$ .

$- 1 - 6 + 2 (- 2 z) - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

Multiply $- 2$ by $2$ .

$- 1 - 6 - 4 z - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

$- 1 - 6 - 4 z - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

Simplify by adding terms.

Subtract $6$ from $- 1$ .

$- 7 - 4 z - 3 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

Subtract $3 z$ from $- 4 z$ .

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 1 + y - z$

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

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

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

Simplify $x = - 1 - 3 - 2 z - z$ .

Simplify the left side.

Remove parentheses.

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

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

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

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

Simplify the right side.

Simplify $- 1 - 3 - 2 z - z$ .

Subtract $3$ from $- 1$ .

$x = - 4 - 2 z - z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

Subtract $z$ from $- 2 z$ .

$x = - 4 - 3 z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 4 - 3 z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 4 - 3 z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 4 - 3 z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

$x = - 4 - 3 z$

$- 7 - 7 z = 7$

$y = - 3 - 2 z$

Solve for $z$ in $- 7 - 7 z = 7$ .

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

Add $7$ to both sides of the equation.

$- 7 z = 7 + 7$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

Add $7$ and $7$ .

$- 7 z = 14$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$- 7 z = 14$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

Divide each term in $- 7 z = 14$ by $- 7$ and simplify.

Divide each term in $- 7 z = 14$ by $- 7$ .

$\frac{- 7 z}{- 7} = \frac{14}{- 7}$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 z}{- 7} = \frac{14}{- 7}$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

Divide $z$ by $1$ .

$z = \frac{14}{- 7}$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$z = \frac{14}{- 7}$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$z = \frac{14}{- 7}$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

Simplify the right side.

Divide $14$ by $- 7$ .

$z = - 2$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$z = - 2$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$z = - 2$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

$z = - 2$

$x = - 4 - 3 z$

$y = - 3 - 2 z$

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

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

$x = - 4 - 3 \cdot - 2$

$z = - 2$

$y = - 3 - 2 z$

Simplify the right side.

Simplify $- 4 - 3 \cdot - 2$ .

Multiply $- 3$ by $- 2$ .

$x = - 4 + 6$

$z = - 2$

$y = - 3 - 2 z$

Add $- 4$ and $6$ .

$x = 2$

$z = - 2$

$y = - 3 - 2 z$

$x = 2$

$z = - 2$

$y = - 3 - 2 z$

$x = 2$

$z = - 2$

$y = - 3 - 2 z$

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

$y = - 3 - 2 \cdot - 2$

$x = 2$

$z = - 2$

Simplify the right side.

Simplify $- 3 - 2 \cdot - 2$ .

Multiply $- 2$ by $- 2$ .

$y = - 3 + 4$

$x = 2$

$z = - 2$

Add $- 3$ and $4$ .

$y = 1$

$x = 2$

$z = - 2$

$y = 1$

$x = 2$

$z = - 2$

$y = 1$

$x = 2$

$z = - 2$

$y = 1$

$x = 2$

$z = - 2$

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

$(2, 1, - 2)$

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 x-y+z=-1 x+y-2z=7 -2x+4y+2z=-4

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