Solve the System of Equations x+y+z=-1 x-y+z=-1 x+y-z=3

$x + y + z = - 1$ $x - y + z = - 1$ $x + y - z = 3$

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

Subtract $y$ from both sides of the equation.

$x + z = - 1 - y$

$x - y + z = - 1$

$x + y - z = 3$

Subtract $z$ from both sides of the equation.

$x = - 1 - y - z$

$x - y + z = - 1$

$x + y - z = 3$

$x = - 1 - y - z$

$x - y + z = - 1$

$x + y - z = 3$

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

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

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

$x = - 1 - y - z$

$x + y - z = 3$

Simplify the left side.

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

Combine the opposite terms in $(- 1 - y - z) - y + z$ .

Add $- z$ and $z$ .

$(- 1 - y) - y + 0 = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

Add $(- 1 - y) - y$ and $0$ .

$(- 1 - y) - y = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

$(- 1 - y) - y = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

Subtract $y$ from $- y$ .

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$x + y - z = 3$

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

$(- 1 - y - z) + y - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Simplify the left side.

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

Combine the opposite terms in $(- 1 - y - z) + y - z$ .

Add $- y$ and $y$ .

$- 1 - z + 0 - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Add $- 1 - z$ and $0$ .

$- 1 - z - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$- 1 - z - z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Subtract $z$ from $- z$ .

$- 1 - 2 z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$- 1 - 2 z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$- 1 - 2 z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$- 1 - 2 z = 3$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Solve for $z$ in $- 1 - 2 z = 3$ .

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

Add $1$ to both sides of the equation.

$- 2 z = 3 + 1$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Add $3$ and $1$ .

$- 2 z = 4$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$- 2 z = 4$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

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

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

$\frac{- 2 z}{- 2} = \frac{4}{- 2}$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 z}{- 2} = \frac{4}{- 2}$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Divide $z$ by $1$ .

$z = \frac{4}{- 2}$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$z = \frac{4}{- 2}$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$z = \frac{4}{- 2}$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

Simplify the right side.

Divide $4$ by $- 2$ .

$z = - 2$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$z = - 2$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$z = - 2$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

$z = - 2$

$- 1 - 2 y = - 1$

$x = - 1 - y - z$

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

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

Add $1$ to both sides of the equation.

$- 2 y = - 1 + 1$

$z = - 2$

$x = - 1 - y - z$

Add $- 1$ and $1$ .

$- 2 y = 0$

$z = - 2$

$x = - 1 - y - z$

$- 2 y = 0$

$z = - 2$

$x = - 1 - y - z$

Divide each term in $- 2 y = 0$ by $- 2$ and simplify.

Divide each term in $- 2 y = 0$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{0}{- 2}$

$z = - 2$

$x = - 1 - y - z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{0}{- 2}$

$z = - 2$

$x = - 1 - y - z$

Divide $y$ by $1$ .

$y = \frac{0}{- 2}$

$z = - 2$

$x = - 1 - y - z$

$y = \frac{0}{- 2}$

$z = - 2$

$x = - 1 - y - z$

$y = \frac{0}{- 2}$

$z = - 2$

$x = - 1 - y - z$

Simplify the right side.

Divide $0$ by $- 2$ .

$y = 0$

$z = - 2$

$x = - 1 - y - z$

$y = 0$

$z = - 2$

$x = - 1 - y - z$

$y = 0$

$z = - 2$

$x = - 1 - y - z$

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

$x = - 1 - y - (- 2)$

$y = 0$

$z = - 2$

Simplify the right side.

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

Multiply $- 1$ by $- 2$ .

$x = - 1 - y + 2$

$y = 0$

$z = - 2$

Add $- 1$ and $2$ .

$x = - y + 1$

$y = 0$

$z = - 2$

$x = - y + 1$

$y = 0$

$z = - 2$

$x = - y + 1$

$y = 0$

$z = - 2$

$x = - y + 1$

$y = 0$

$z = - 2$

Replace all occurrences of $y$ with $0$ in each equation.

Replace all occurrences of $y$ in $x = - y + 1$ with $0$ .

$x = - (0) + 1$

$y = 0$

$z = - 2$

Simplify the right side.

Simplify $- (0) + 1$ .

Multiply $- 1$ by $0$ .

$x = 0 + 1$

$y = 0$

$z = - 2$

Add $0$ and $1$ .

$x = 1$

$y = 0$

$z = - 2$

$x = 1$

$y = 0$

$z = - 2$

$x = 1$

$y = 0$

$z = - 2$

$x = 1$

$y = 0$

$z = - 2$

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

$(1, 0, - 2)$

The result can be shown in multiple forms.

Point Form:

$(1, 0, - 2)$

Equation Form:

$x = 1, y = 0, z = - 2$

Solve the System of Equations x+y+z=-1 x-y+z=-1 x+y-z=3

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