Solve the System of Equations -9x-y+2z=-4 -6x+y+5z=2 -4x+y+5z=6

$- 9 x - y + 2 z = - 4$ $- 6 x + y + 5 z = 2$ $- 4 x + y + 5 z = 6$

Step 1

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

Add $6 x$ to both sides of the equation.

$y + 5 z = 2 + 6 x$

$- 9 x - y + 2 z = - 4$

$- 4 x + y + 5 z = 6$

Subtract $5 z$ from both sides of the equation.

$y = 2 + 6 x - 5 z$

$- 9 x - y + 2 z = - 4$

$- 4 x + y + 5 z = 6$

$y = 2 + 6 x - 5 z$

$- 9 x - y + 2 z = - 4$

$- 4 x + y + 5 z = 6$

Step 2

Replace all occurrences of $y$ with $2 + 6 x - 5 z$ in each equation.

Replace all occurrences of $y$ in $- 9 x - y + 2 z = - 4$ with $2 + 6 x - 5 z$ .

$- 9 x - (2 + 6 x - 5 z) + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Simplify the left side.

Simplify $- 9 x - (2 + 6 x - 5 z) + 2 z$ .

Simplify each term.

Apply the distributive property.

$- 9 x - 1 \cdot 2 - (6 x) - (- 5 z) + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Simplify.

Multiply $- 1$ by $2$ .

$- 9 x - 2 - (6 x) - (- 5 z) + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Multiply $6$ by $- 1$ .

$- 9 x - 2 - 6 x - (- 5 z) + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Multiply $- 5$ by $- 1$ .

$- 9 x - 2 - 6 x + 5 z + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

$- 9 x - 2 - 6 x + 5 z + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

$- 9 x - 2 - 6 x + 5 z + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Simplify by adding terms.

Subtract $6 x$ from $- 9 x$ .

$- 15 x - 2 + 5 z + 2 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Add $5 z$ and $2 z$ .

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + y + 5 z = 6$

Replace all occurrences of $y$ in $- 4 x + y + 5 z = 6$ with $2 + 6 x - 5 z$ .

$- 4 x + 2 + 6 x - 5 z + 5 z = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Simplify the left side.

Simplify $- 4 x + 2 + 6 x - 5 z + 5 z$ .

Remove parentheses.

$- 4 x + 2 + 6 x - 5 z + 5 z = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Combine the opposite terms in $- 4 x + 2 + 6 x - 5 z + 5 z$ .

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

$- 4 x + 2 + 6 x + 0 = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Add $- 4 x + 2 + 6 x$ and $0$ .

$- 4 x + 2 + 6 x = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$- 4 x + 2 + 6 x = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Add $- 4 x$ and $6 x$ .

$2 x + 2 = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$2 x + 2 = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$2 x + 2 = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$2 x + 2 = 6$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Step 3

Solve for $x$ in $2 x + 2 = 6$ .

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

Subtract $2$ from both sides of the equation.

$2 x = 6 - 2$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Subtract $2$ from $6$ .

$2 x = 4$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$2 x = 4$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

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

Divide each term in $2 x = 4$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2}$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2}$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Divide $x$ by $1$ .

$x = \frac{4}{2}$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$x = \frac{4}{2}$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$x = \frac{4}{2}$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Simplify the right side.

Divide $4$ by $2$ .

$x = 2$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$x = 2$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$x = 2$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

$x = 2$

$- 15 x - 2 + 7 z = - 4$

$y = 2 + 6 x - 5 z$

Step 4

Replace all occurrences of $x$ with $2$ in each equation.

Replace all occurrences of $x$ in $- 15 x - 2 + 7 z = - 4$ with $2$ .

$- 15 \cdot 2 - 2 + 7 z = - 4$

$x = 2$

$y = 2 + 6 x - 5 z$

Simplify the left side.

Simplify $- 15 \cdot 2 - 2 + 7 z$ .

Multiply $- 15$ by $2$ .

$- 30 - 2 + 7 z = - 4$

$x = 2$

$y = 2 + 6 x - 5 z$

Subtract $2$ from $- 30$ .

$- 32 + 7 z = - 4$

$x = 2$

$y = 2 + 6 x - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

$y = 2 + 6 x - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

$y = 2 + 6 x - 5 z$

Replace all occurrences of $x$ in $y = 2 + 6 x - 5 z$ with $2$ .

$y = 2 + 6 (2) - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

Simplify the right side.

Simplify $2 + 6 (2) - 5 z$ .

Multiply $6$ by $2$ .

$y = 2 + 12 - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

Add $2$ and $12$ .

$y = 14 - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

$y = 14 - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

$y = 14 - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

$y = 14 - 5 z$

$- 32 + 7 z = - 4$

$x = 2$

Step 5

Solve for $z$ in $- 32 + 7 z = - 4$ .

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

Add $32$ to both sides of the equation.

$7 z = - 4 + 32$

$y = 14 - 5 z$

$x = 2$

Add $- 4$ and $32$ .

$7 z = 28$

$y = 14 - 5 z$

$x = 2$

$7 z = 28$

$y = 14 - 5 z$

$x = 2$

Divide each term in $7 z = 28$ by $7$ and simplify.

Divide each term in $7 z = 28$ by $7$ .

$\frac{7 z}{7} = \frac{28}{7}$

$y = 14 - 5 z$

$x = 2$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 z}{7} = \frac{28}{7}$

$y = 14 - 5 z$

$x = 2$

Divide $z$ by $1$ .

$z = \frac{28}{7}$

$y = 14 - 5 z$

$x = 2$

$z = \frac{28}{7}$

$y = 14 - 5 z$

$x = 2$

$z = \frac{28}{7}$

$y = 14 - 5 z$

$x = 2$

Simplify the right side.

Divide $28$ by $7$ .

$z = 4$

$y = 14 - 5 z$

$x = 2$

$z = 4$

$y = 14 - 5 z$

$x = 2$

$z = 4$

$y = 14 - 5 z$

$x = 2$

$z = 4$

$y = 14 - 5 z$

$x = 2$

Step 6

Replace all occurrences of $z$ with $4$ in each equation.

Replace all occurrences of $z$ in $y = 14 - 5 z$ with $4$ .

$y = 14 - 5 \cdot 4$

$z = 4$

$x = 2$

Simplify the right side.

Simplify $14 - 5 \cdot 4$ .

Multiply $- 5$ by $4$ .

$y = 14 - 20$

$z = 4$

$x = 2$

Subtract $20$ from $14$ .

$y = - 6$

$z = 4$

$x = 2$

$y = - 6$

$z = 4$

$x = 2$

$y = - 6$

$z = 4$

$x = 2$

$y = - 6$

$z = 4$

$x = 2$

Step 7

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

$(2, - 6, 4)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, - 6, 4)$

Equation Form:

$x = 2, y = - 6, z = 4$

Solve the System of Equations -9x-y+2z=-4 -6x+y+5z=2 -4x+y+5z=6

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