Solve the System of Equations -6x+6y-7z=-7 6x+y+6z=-7 6x+y+8z=7

$- 6 x + 6 y - 7 z = - 7$ $6 x + y + 6 z = - 7$ $6 x + y + 8 z = 7$

Step 1

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

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

$y + 6 z = - 7 - 6 x$

$- 6 x + 6 y - 7 z = - 7$

$6 x + y + 8 z = 7$

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

$y = - 7 - 6 x - 6 z$

$- 6 x + 6 y - 7 z = - 7$

$6 x + y + 8 z = 7$

$y = - 7 - 6 x - 6 z$

$- 6 x + 6 y - 7 z = - 7$

$6 x + y + 8 z = 7$

Step 2

Replace all occurrences of $y$ with $- 7 - 6 x - 6 z$ in each equation.

Replace all occurrences of $y$ in $- 6 x + 6 y - 7 z = - 7$ with $- 7 - 6 x - 6 z$ .

$- 6 x + 6 (- 7 - 6 x - 6 z) - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Simplify the left side.

Simplify $- 6 x + 6 (- 7 - 6 x - 6 z) - 7 z$ .

Simplify each term.

Apply the distributive property.

$- 6 x + 6 \cdot - 7 + 6 (- 6 x) + 6 (- 6 z) - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Simplify.

Multiply $6$ by $- 7$ .

$- 6 x - 42 + 6 (- 6 x) + 6 (- 6 z) - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Multiply $- 6$ by $6$ .

$- 6 x - 42 - 36 x + 6 (- 6 z) - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Multiply $- 6$ by $6$ .

$- 6 x - 42 - 36 x - 36 z - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

$- 6 x - 42 - 36 x - 36 z - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

$- 6 x - 42 - 36 x - 36 z - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Simplify by adding terms.

Subtract $36 x$ from $- 6 x$ .

$- 42 x - 42 - 36 z - 7 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Subtract $7 z$ from $- 36 z$ .

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$6 x + y + 8 z = 7$

Replace all occurrences of $y$ in $6 x + y + 8 z = 7$ with $- 7 - 6 x - 6 z$ .

$6 x - 7 - 6 x - 6 z + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Simplify the left side.

Simplify $6 x - 7 - 6 x - 6 z + 8 z$ .

Remove parentheses.

$6 x - 7 - 6 x - 6 z + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Combine the opposite terms in $6 x - 7 - 6 x - 6 z + 8 z$ .

Subtract $6 x$ from $6 x$ .

$0 - 7 - 6 z + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Subtract $7$ from $0$ .

$- 7 - 6 z + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$- 7 - 6 z + 8 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Add $- 6 z$ and $8 z$ .

$- 7 + 2 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$- 7 + 2 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$- 7 + 2 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$- 7 + 2 z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Step 3

Solve for $z$ in $- 7 + 2 z = 7$ .

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

Add $7$ to both sides of the equation.

$2 z = 7 + 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Add $7$ and $7$ .

$2 z = 14$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$2 z = 14$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Divide each term in $2 z = 14$ by $2$ and simplify.

Divide each term in $2 z = 14$ by $2$ .

$\frac{2 z}{2} = \frac{14}{2}$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{14}{2}$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Divide $z$ by $1$ .

$z = \frac{14}{2}$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$z = \frac{14}{2}$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$z = \frac{14}{2}$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Simplify the right side.

Divide $14$ by $2$ .

$z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

$z = 7$

$- 42 x - 42 - 43 z = - 7$

$y = - 7 - 6 x - 6 z$

Step 4

Replace all occurrences of $z$ with $7$ in each equation.

Replace all occurrences of $z$ in $- 42 x - 42 - 43 z = - 7$ with $7$ .

$- 42 x - 42 - 43 \cdot 7 = - 7$

$z = 7$

$y = - 7 - 6 x - 6 z$

Simplify the left side.

Simplify $- 42 x - 42 - 43 \cdot 7$ .

Multiply $- 43$ by $7$ .

$- 42 x - 42 - 301 = - 7$

$z = 7$

$y = - 7 - 6 x - 6 z$

Subtract $301$ from $- 42$ .

$- 42 x - 343 = - 7$

$z = 7$

$y = - 7 - 6 x - 6 z$

$- 42 x - 343 = - 7$

$z = 7$

$y = - 7 - 6 x - 6 z$

$- 42 x - 343 = - 7$

$z = 7$

$y = - 7 - 6 x - 6 z$

Replace all occurrences of $z$ in $y = - 7 - 6 x - 6 z$ with $7$ .

$y = - 7 - 6 x - 6 \cdot 7$

$- 42 x - 343 = - 7$

$z = 7$

Simplify the right side.

Simplify $- 7 - 6 x - 6 \cdot 7$ .

Multiply $- 6$ by $7$ .

$y = - 7 - 6 x - 42$

$- 42 x - 343 = - 7$

$z = 7$

Subtract $42$ from $- 7$ .

$y = - 6 x - 49$

$- 42 x - 343 = - 7$

$z = 7$

$y = - 6 x - 49$

$- 42 x - 343 = - 7$

$z = 7$

$y = - 6 x - 49$

$- 42 x - 343 = - 7$

$z = 7$

$y = - 6 x - 49$

$- 42 x - 343 = - 7$

$z = 7$

Step 5

Solve for $x$ in $- 42 x - 343 = - 7$ .

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

Add $343$ to both sides of the equation.

$- 42 x = - 7 + 343$

$y = - 6 x - 49$

$z = 7$

Add $- 7$ and $343$ .

$- 42 x = 336$

$y = - 6 x - 49$

$z = 7$

$- 42 x = 336$

$y = - 6 x - 49$

$z = 7$

Divide each term in $- 42 x = 336$ by $- 42$ and simplify.

Divide each term in $- 42 x = 336$ by $- 42$ .

$\frac{- 42 x}{- 42} = \frac{336}{- 42}$

$y = - 6 x - 49$

$z = 7$

Simplify the left side.

Cancel the common factor of $- 42$ .

Cancel the common factor.

$\frac{- 42 x}{- 42} = \frac{336}{- 42}$

$y = - 6 x - 49$

$z = 7$

Divide $x$ by $1$ .

$x = \frac{336}{- 42}$

$y = - 6 x - 49$

$z = 7$

$x = \frac{336}{- 42}$

$y = - 6 x - 49$

$z = 7$

$x = \frac{336}{- 42}$

$y = - 6 x - 49$

$z = 7$

Simplify the right side.

Divide $336$ by $- 42$ .

$x = - 8$

$y = - 6 x - 49$

$z = 7$

$x = - 8$

$y = - 6 x - 49$

$z = 7$

$x = - 8$

$y = - 6 x - 49$

$z = 7$

$x = - 8$

$y = - 6 x - 49$

$z = 7$

Step 6

Replace all occurrences of $x$ with $- 8$ in each equation.

Replace all occurrences of $x$ in $y = - 6 x - 49$ with $- 8$ .

$y = - 6 \cdot - 8 - 49$

$x = - 8$

$z = 7$

Simplify the right side.

Simplify $- 6 \cdot - 8 - 49$ .

Multiply $- 6$ by $- 8$ .

$y = 48 - 49$

$x = - 8$

$z = 7$

Subtract $49$ from $48$ .

$y = - 1$

$x = - 8$

$z = 7$

$y = - 1$

$x = - 8$

$z = 7$

$y = - 1$

$x = - 8$

$z = 7$

$y = - 1$

$x = - 8$

$z = 7$

Step 7

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

$(- 8, - 1, 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 8, - 1, 7)$

Equation Form:

$x = - 8, y = - 1, z = 7$

Solve the System of Equations -6x+6y-7z=-7 6x+y+6z=-7 6x+y+8z=7

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