Solve the System of Equations x+y+z=9 x-y-z=-1 x-y+z=21

$x + y + z = 9$ $x - y - z = - 1$ $x - y + z = 21$

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

Subtract $y$ from both sides of the equation.

$x + z = 9 - y$

$x - y - z = - 1$

$x - y + z = 21$

Subtract $z$ from both sides of the equation.

$x = 9 - y - z$

$x - y - z = - 1$

$x - y + z = 21$

$x = 9 - y - z$

$x - y - z = - 1$

$x - y + z = 21$

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

Replace all occurrences of $x$ in $x - y - z = - 1$ with $9 - y - z$ .

$(9 - y - z) - y - z = - 1$

$x = 9 - y - z$

$x - y + z = 21$

Simplify the left side.

Simplify $(9 - y - z) - y - z$ .

Subtract $y$ from $- y$ .

$9 - 2 y - z - z = - 1$

$x = 9 - y - z$

$x - y + z = 21$

Subtract $z$ from $- z$ .

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$x - y + z = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$x - y + z = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$x - y + z = 21$

Replace all occurrences of $x$ in $x - y + z = 21$ with $9 - y - z$ .

$(9 - y - z) - y + z = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Simplify the left side.

Simplify $(9 - y - z) - y + z$ .

Combine the opposite terms in $(9 - y - z) - y + z$ .

Add $- z$ and $z$ .

$(9 - y) - y + 0 = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Add $(9 - y) - y$ and $0$ .

$(9 - y) - y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$(9 - y) - y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Subtract $y$ from $- y$ .

$9 - 2 y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$9 - 2 y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$9 - 2 y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$9 - 2 y = 21$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Solve for $y$ in $9 - 2 y = 21$ .

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

Subtract $9$ from both sides of the equation.

$- 2 y = 21 - 9$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Subtract $9$ from $21$ .

$- 2 y = 12$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$- 2 y = 12$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Divide each term in $- 2 y = 12$ by $- 2$ and simplify.

Divide each term in $- 2 y = 12$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{12}{- 2}$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{12}{- 2}$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Divide $y$ by $1$ .

$y = \frac{12}{- 2}$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$y = \frac{12}{- 2}$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$y = \frac{12}{- 2}$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Simplify the right side.

Divide $12$ by $- 2$ .

$y = - 6$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$y = - 6$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$y = - 6$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

$y = - 6$

$9 - 2 y - 2 z = - 1$

$x = 9 - y - z$

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $9 - 2 y - 2 z = - 1$ with $- 6$ .

$9 - 2 \cdot - 6 - 2 z = - 1$

$y = - 6$

$x = 9 - y - z$

Simplify the left side.

Simplify $9 - 2 \cdot - 6 - 2 z$ .

Multiply $- 2$ by $- 6$ .

$9 + 12 - 2 z = - 1$

$y = - 6$

$x = 9 - y - z$

Add $9$ and $12$ .

$21 - 2 z = - 1$

$y = - 6$

$x = 9 - y - z$

$21 - 2 z = - 1$

$y = - 6$

$x = 9 - y - z$

$21 - 2 z = - 1$

$y = - 6$

$x = 9 - y - z$

Replace all occurrences of $y$ in $x = 9 - y - z$ with $- 6$ .

$x = 9 - (- 6) - z$

$21 - 2 z = - 1$

$y = - 6$

Simplify the right side.

Simplify $9 - (- 6) - z$ .

Multiply $- 1$ by $- 6$ .

$x = 9 + 6 - z$

$21 - 2 z = - 1$

$y = - 6$

Add $9$ and $6$ .

$x = 15 - z$

$21 - 2 z = - 1$

$y = - 6$

$x = 15 - z$

$21 - 2 z = - 1$

$y = - 6$

$x = 15 - z$

$21 - 2 z = - 1$

$y = - 6$

$x = 15 - z$

$21 - 2 z = - 1$

$y = - 6$

Solve for $z$ in $21 - 2 z = - 1$ .

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

Subtract $21$ from both sides of the equation.

$- 2 z = - 1 - 21$

$x = 15 - z$

$y = - 6$

Subtract $21$ from $- 1$ .

$- 2 z = - 22$

$x = 15 - z$

$y = - 6$

$- 2 z = - 22$

$x = 15 - z$

$y = - 6$

Divide each term in $- 2 z = - 22$ by $- 2$ and simplify.

Divide each term in $- 2 z = - 22$ by $- 2$ .

$\frac{- 2 z}{- 2} = \frac{- 22}{- 2}$

$x = 15 - z$

$y = - 6$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 z}{- 2} = \frac{- 22}{- 2}$

$x = 15 - z$

$y = - 6$

Divide $z$ by $1$ .

$z = \frac{- 22}{- 2}$

$x = 15 - z$

$y = - 6$

$z = \frac{- 22}{- 2}$

$x = 15 - z$

$y = - 6$

$z = \frac{- 22}{- 2}$

$x = 15 - z$

$y = - 6$

Simplify the right side.

Divide $- 22$ by $- 2$ .

$z = 11$

$x = 15 - z$

$y = - 6$

$z = 11$

$x = 15 - z$

$y = - 6$

$z = 11$

$x = 15 - z$

$y = - 6$

$z = 11$

$x = 15 - z$

$y = - 6$

Replace all occurrences of $z$ with $11$ in each equation.

Replace all occurrences of $z$ in $x = 15 - z$ with $11$ .

$x = 15 - (11)$

$z = 11$

$y = - 6$

Simplify the right side.

Simplify $15 - (11)$ .

Multiply $- 1$ by $11$ .

$x = 15 - 11$

$z = 11$

$y = - 6$

Subtract $11$ from $15$ .

$x = 4$

$z = 11$

$y = - 6$

$x = 4$

$z = 11$

$y = - 6$

$x = 4$

$z = 11$

$y = - 6$

$x = 4$

$z = 11$

$y = - 6$

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

$(4, - 6, 11)$

The result can be shown in multiple forms.

Point Form:

$(4, - 6, 11)$

Equation Form:

$x = 4, y = - 6, z = 11$

Solve the System of Equations x+y+z=9 x-y-z=-1 x-y+z=21

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