Solve the System of Equations -x-3y-2z=8 -x+y+6z=4 x-9y-2z=4

$- x - 3 y - 2 z = 8$ $- x + y + 6 z = 4$ $x - 9 y - 2 z = 4$

Step 1

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

Add $x$ to both sides of the equation.

$y + 6 z = 4 + x$

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

$x - 9 y - 2 z = 4$

Subtract $6 z$ from both sides of the equation.

$y = 4 + x - 6 z$

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

$x - 9 y - 2 z = 4$

$y = 4 + x - 6 z$

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

$x - 9 y - 2 z = 4$

Step 2

Replace all occurrences of $y$ with $4 + x - 6 z$ in each equation.

Replace all occurrences of $y$ in $- x - 3 y - 2 z = 8$ with $4 + x - 6 z$ .

$- x - 3 (4 + x - 6 z) - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Simplify the left side.

Simplify $- x - 3 (4 + x - 6 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$- x - 3 \cdot 4 - 3 x - 3 (- 6 z) - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Simplify.

Multiply $- 3$ by $4$ .

$- x - 12 - 3 x - 3 (- 6 z) - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Multiply $- 6$ by $- 3$ .

$- x - 12 - 3 x + 18 z - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

$- x - 12 - 3 x + 18 z - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

$- x - 12 - 3 x + 18 z - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Simplify by adding terms.

Subtract $3 x$ from $- x$ .

$- 4 x - 12 + 18 z - 2 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Subtract $2 z$ from $18 z$ .

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 9 y - 2 z = 4$

Replace all occurrences of $y$ in $x - 9 y - 2 z = 4$ with $4 + x - 6 z$ .

$x - 9 (4 + x - 6 z) - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Simplify the left side.

Simplify $x - 9 (4 + x - 6 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$x - 9 \cdot 4 - 9 x - 9 (- 6 z) - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Simplify.

Multiply $- 9$ by $4$ .

$x - 36 - 9 x - 9 (- 6 z) - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Multiply $- 6$ by $- 9$ .

$x - 36 - 9 x + 54 z - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 36 - 9 x + 54 z - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x - 36 - 9 x + 54 z - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Simplify by adding terms.

Subtract $9 x$ from $x$ .

$- 8 x - 36 + 54 z - 2 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Subtract $2 z$ from $54 z$ .

$- 8 x - 36 + 52 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$- 8 x - 36 + 52 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$- 8 x - 36 + 52 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$- 8 x - 36 + 52 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$- 8 x - 36 + 52 z = 4$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Step 3

Solve for $x$ in $- 8 x - 36 + 52 z = 4$ .

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

Add $36$ to both sides of the equation.

$- 8 x + 52 z = 4 + 36$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Subtract $52 z$ from both sides of the equation.

$- 8 x = 4 + 36 - 52 z$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Add $4$ and $36$ .

$- 8 x = 40 - 52 z$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$- 8 x = 40 - 52 z$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Divide each term in $- 8 x = 40 - 52 z$ by $- 8$ and simplify.

Divide each term in $- 8 x = 40 - 52 z$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{40}{- 8} + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{40}{- 8} + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Divide $x$ by $1$ .

$x = \frac{40}{- 8} + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = \frac{40}{- 8} + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = \frac{40}{- 8} + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Simplify the right side.

Simplify each term.

Divide $40$ by $- 8$ .

$x = - 5 + \frac{- 52 z}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Cancel the common factor of $- 52$ and $- 8$ .

Factor $- 4$ out of $- 52 z$ .

$x = - 5 + \frac{- 4 (13 z)}{- 8}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Cancel the common factors.

Factor $- 4$ out of $- 8$ .

$x = - 5 + \frac{- 4 (13 z)}{- 4 \cdot 2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Cancel the common factor.

$x = - 5 + \frac{- 4 (13 z)}{- 4 \cdot 2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Rewrite the expression.

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

$x = - 5 + \frac{13 z}{2}$

$- 4 x - 12 + 16 z = 8$

$y = 4 + x - 6 z$

Step 4

Replace all occurrences of $x$ with $- 5 + \frac{13 z}{2}$ in each equation.

Replace all occurrences of $x$ in $- 4 x - 12 + 16 z = 8$ with $- 5 + \frac{13 z}{2}$ .

$- 4 (- 5 + \frac{13 z}{2}) - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Simplify the left side.

Simplify $- 4 (- 5 + \frac{13 z}{2}) - 12 + 16 z$ .

Simplify each term.

Apply the distributive property.

$- 4 \cdot - 5 - 4 \frac{13 z}{2} - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Multiply $- 4$ by $- 5$ .

$20 - 4 \frac{13 z}{2} - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

$20 + 2 (- 2) (\frac{13 z}{2}) - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Cancel the common factor.

$20 + 2 \cdot (- 2 \frac{13 z}{2}) - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Rewrite the expression.

$20 - 2 (13 z) - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

$20 - 2 (13 z) - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Multiply $13$ by $- 2$ .

$20 - 26 z - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

$20 - 26 z - 12 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Simplify by adding terms.

Subtract $12$ from $20$ .

$- 26 z + 8 + 16 z = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Add $- 26 z$ and $16 z$ .

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 + x - 6 z$

Replace all occurrences of $x$ in $y = 4 + x - 6 z$ with $- 5 + \frac{13 z}{2}$ .

$y = 4 - 5 + \frac{13 z}{2} - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Simplify $y = 4 - 5 + \frac{13 z}{2} - 6 z$ .

Simplify the left side.

Remove parentheses.

$y = 4 - 5 + \frac{13 z}{2} - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = 4 - 5 + \frac{13 z}{2} - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Simplify the right side.

Simplify $4 - 5 + \frac{13 z}{2} - 6 z$ .

Subtract $5$ from $4$ .

$y = - 1 + \frac{13 z}{2} - 6 z$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

To write $- 6 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - 1 + \frac{13 z}{2} - 6 z \cdot \frac{2}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Simplify terms.

Combine $- 6 z$ and $\frac{2}{2}$ .

$y = - 1 + \frac{13 z}{2} + \frac{- 6 z \cdot 2}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Combine the numerators over the common denominator.

$y = - 1 + \frac{13 z - 6 z \cdot 2}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = - 1 + \frac{13 z - 6 z \cdot 2}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Simplify each term.

Simplify the numerator.

Factor $z$ out of $13 z - 6 z \cdot 2$ .

Factor $z$ out of $13 z$ .

$y = - 1 + \frac{z \cdot 13 - 6 z \cdot 2}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Factor $z$ out of $- 6 z \cdot 2$ .

$y = - 1 + \frac{z \cdot 13 + z (- 6 \cdot 2)}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Factor $z$ out of $z \cdot 13 + z (- 6 \cdot 2)$ .

$y = - 1 + \frac{z (13 - 6 \cdot 2)}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

$y = - 1 + \frac{z (13 - 6 \cdot 2)}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Multiply $- 6$ by $2$ .

$y = - 1 + \frac{z (13 - 12)}{2}$

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Subtract $12$ from $13$ .

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Multiply $z$ by $1$ .

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

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

$- 10 z + 8 = 8$

$x = - 5 + \frac{13 z}{2}$

Step 5

Solve for $z$ in $- 10 z + 8 = 8$ .

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

Subtract $8$ from both sides of the equation.

$- 10 z = 8 - 8$

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

$x = - 5 + \frac{13 z}{2}$

Subtract $8$ from $8$ .

$- 10 z = 0$

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

$x = - 5 + \frac{13 z}{2}$

$- 10 z = 0$

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

$x = - 5 + \frac{13 z}{2}$

Divide each term in $- 10 z = 0$ by $- 10$ and simplify.

Divide each term in $- 10 z = 0$ by $- 10$ .

$\frac{- 10 z}{- 10} = \frac{0}{- 10}$

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

$x = - 5 + \frac{13 z}{2}$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 z}{- 10} = \frac{0}{- 10}$

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

$x = - 5 + \frac{13 z}{2}$

Divide $z$ by $1$ .

$z = \frac{0}{- 10}$

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

$x = - 5 + \frac{13 z}{2}$

$z = \frac{0}{- 10}$

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

$x = - 5 + \frac{13 z}{2}$

$z = \frac{0}{- 10}$

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

$x = - 5 + \frac{13 z}{2}$

Simplify the right side.

Divide $0$ by $- 10$ .

$z = 0$

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

$x = - 5 + \frac{13 z}{2}$

$z = 0$

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

$x = - 5 + \frac{13 z}{2}$

$z = 0$

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

$x = - 5 + \frac{13 z}{2}$

$z = 0$

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

$x = - 5 + \frac{13 z}{2}$

Step 6

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

Replace all occurrences of $z$ in $y = - 1 + \frac{z}{2}$ with $0$ .

$y = - 1 + \frac{0}{2}$

$z = 0$

$x = - 5 + \frac{13 z}{2}$

Simplify the right side.

Simplify $- 1 + \frac{0}{2}$ .

Divide $0$ by $2$ .

$y = - 1 + 0$

$z = 0$

$x = - 5 + \frac{13 z}{2}$

Add $- 1$ and $0$ .

$y = - 1$

$z = 0$

$x = - 5 + \frac{13 z}{2}$

$y = - 1$

$z = 0$

$x = - 5 + \frac{13 z}{2}$

$y = - 1$

$z = 0$

$x = - 5 + \frac{13 z}{2}$

Replace all occurrences of $z$ in $x = - 5 + \frac{13 z}{2}$ with $0$ .

$x = - 5 + \frac{13 (0)}{2}$

$y = - 1$

$z = 0$

Simplify the right side.

Simplify $- 5 + \frac{13 (0)}{2}$ .

Multiply $13$ by $0$ .

$x = - 5 + \frac{0}{2}$

$y = - 1$

$z = 0$

Divide $0$ by $2$ .

$x = - 5 + 0$

$y = - 1$

$z = 0$

Add $- 5$ and $0$ .

$x = - 5$

$y = - 1$

$z = 0$

$x = - 5$

$y = - 1$

$z = 0$

$x = - 5$

$y = - 1$

$z = 0$

$x = - 5$

$y = - 1$

$z = 0$

Step 7

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

$(- 5, - 1, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 5, - 1, 0)$

Equation Form:

$x = - 5, y = - 1, z = 0$

Solve the System of Equations -x-3y-2z=8 -x+y+6z=4 x-9y-2z=4

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