Solve the System of Equations x+2y-6z=4 x-3y+9z=9 -x-3y+7z=-1

$x + 2 y - 6 z = 4$ $x - 3 y + 9 z = 9$ $- x - 3 y + 7 z = - 1$

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

Subtract $2 y$ from both sides of the equation.

$x - 6 z = 4 - 2 y$

$x - 3 y + 9 z = 9$

$- x - 3 y + 7 z = - 1$

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

$x = 4 - 2 y + 6 z$

$x - 3 y + 9 z = 9$

$- x - 3 y + 7 z = - 1$

$x = 4 - 2 y + 6 z$

$x - 3 y + 9 z = 9$

$- x - 3 y + 7 z = - 1$

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

Replace all occurrences of $x$ in $x - 3 y + 9 z = 9$ with $4 - 2 y + 6 z$ .

$(4 - 2 y + 6 z) - 3 y + 9 z = 9$

$x = 4 - 2 y + 6 z$

$- x - 3 y + 7 z = - 1$

Simplify the left side.

Simplify $(4 - 2 y + 6 z) - 3 y + 9 z$ .

Subtract $3 y$ from $- 2 y$ .

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

$x = 4 - 2 y + 6 z$

$- x - 3 y + 7 z = - 1$

Add $6 z$ and $9 z$ .

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- x - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- x - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- x - 3 y + 7 z = - 1$

Replace all occurrences of $x$ in $- x - 3 y + 7 z = - 1$ with $4 - 2 y + 6 z$ .

$- (4 - 2 y + 6 z) - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Simplify the left side.

Simplify $- (4 - 2 y + 6 z) - 3 y + 7 z$ .

Simplify each term.

Apply the distributive property.

$- 1 \cdot 4 - (- 2 y) - (6 z) - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Simplify.

Multiply $- 1$ by $4$ .

$- 4 - (- 2 y) - (6 z) - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Multiply $- 2$ by $- 1$ .

$- 4 + 2 y - (6 z) - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Multiply $6$ by $- 1$ .

$- 4 + 2 y - 6 z - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 + 2 y - 6 z - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 + 2 y - 6 z - 3 y + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Simplify by adding terms.

Subtract $3 y$ from $2 y$ .

$- 4 - y - 6 z + 7 z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Add $- 6 z$ and $7 z$ .

$- 4 - y + z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 - y + z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 - y + z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 - y + z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$- 4 - y + z = - 1$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

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

Add $4$ to both sides of the equation.

$- y + z = - 1 + 4$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Add $y$ to both sides of the equation.

$z = - 1 + 4 + y$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Add $- 1$ and $4$ .

$z = 3 + y$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

$z = 3 + y$

$4 - 5 y + 15 z = 9$

$x = 4 - 2 y + 6 z$

Replace all occurrences of $z$ with $3 + y$ in each equation.

Replace all occurrences of $z$ in $4 - 5 y + 15 z = 9$ with $3 + y$ .

$4 - 5 y + 15 (3 + y) = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

Simplify the left side.

Simplify $4 - 5 y + 15 (3 + y)$ .

Simplify each term.

Apply the distributive property.

$4 - 5 y + 15 \cdot 3 + 15 y = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

Multiply $15$ by $3$ .

$4 - 5 y + 45 + 15 y = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

$4 - 5 y + 45 + 15 y = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

Simplify by adding terms.

Add $4$ and $45$ .

$- 5 y + 49 + 15 y = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

Add $- 5 y$ and $15 y$ .

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 - 2 y + 6 z$

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

$x = 4 - 2 y + 6 (3 + y)$

$10 y + 49 = 9$

$z = 3 + y$

Simplify the right side.

Simplify $4 - 2 y + 6 (3 + y)$ .

Simplify each term.

Apply the distributive property.

$x = 4 - 2 y + 6 \cdot 3 + 6 y$

$10 y + 49 = 9$

$z = 3 + y$

Multiply $6$ by $3$ .

$x = 4 - 2 y + 18 + 6 y$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 - 2 y + 18 + 6 y$

$10 y + 49 = 9$

$z = 3 + y$

Simplify by adding terms.

Add $4$ and $18$ .

$x = - 2 y + 22 + 6 y$

$10 y + 49 = 9$

$z = 3 + y$

Add $- 2 y$ and $6 y$ .

$x = 4 y + 22$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 y + 22$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 y + 22$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 y + 22$

$10 y + 49 = 9$

$z = 3 + y$

$x = 4 y + 22$

$10 y + 49 = 9$

$z = 3 + y$

Solve for $y$ in $10 y + 49 = 9$ .

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

Subtract $49$ from both sides of the equation.

$10 y = 9 - 49$

$x = 4 y + 22$

$z = 3 + y$

Subtract $49$ from $9$ .

$10 y = - 40$

$x = 4 y + 22$

$z = 3 + y$

$10 y = - 40$

$x = 4 y + 22$

$z = 3 + y$

Divide each term in $10 y = - 40$ by $10$ and simplify.

Divide each term in $10 y = - 40$ by $10$ .

$\frac{10 y}{10} = \frac{- 40}{10}$

$x = 4 y + 22$

$z = 3 + y$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 y}{10} = \frac{- 40}{10}$

$x = 4 y + 22$

$z = 3 + y$

Divide $y$ by $1$ .

$y = \frac{- 40}{10}$

$x = 4 y + 22$

$z = 3 + y$

$y = \frac{- 40}{10}$

$x = 4 y + 22$

$z = 3 + y$

$y = \frac{- 40}{10}$

$x = 4 y + 22$

$z = 3 + y$

Simplify the right side.

Divide $- 40$ by $10$ .

$y = - 4$

$x = 4 y + 22$

$z = 3 + y$

$y = - 4$

$x = 4 y + 22$

$z = 3 + y$

$y = - 4$

$x = 4 y + 22$

$z = 3 + y$

$y = - 4$

$x = 4 y + 22$

$z = 3 + y$

Replace all occurrences of $y$ with $- 4$ in each equation.

Replace all occurrences of $y$ in $x = 4 y + 22$ with $- 4$ .

$x = 4 (- 4) + 22$

$y = - 4$

$z = 3 + y$

Simplify the right side.

Simplify $4 (- 4) + 22$ .

Multiply $4$ by $- 4$ .

$x = - 16 + 22$

$y = - 4$

$z = 3 + y$

Add $- 16$ and $22$ .

$x = 6$

$y = - 4$

$z = 3 + y$

$x = 6$

$y = - 4$

$z = 3 + y$

$x = 6$

$y = - 4$

$z = 3 + y$

Replace all occurrences of $y$ in $z = 3 + y$ with $- 4$ .

$z = 3 - 4$

$x = 6$

$y = - 4$

Simplify $z = 3 - 4$ .

Simplify the left side.

Remove parentheses.

$z = 3 - 4$

$x = 6$

$y = - 4$

$z = 3 - 4$

$x = 6$

$y = - 4$

Simplify the right side.

Subtract $4$ from $3$ .

$z = - 1$

$x = 6$

$y = - 4$

$z = - 1$

$x = 6$

$y = - 4$

$z = - 1$

$x = 6$

$y = - 4$

$z = - 1$

$x = 6$

$y = - 4$

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

$(6, - 4, - 1)$

The result can be shown in multiple forms.

Point Form:

$(6, - 4, - 1)$

Equation Form:

$x = 6, y = - 4, z = - 1$

Solve the System of Equations x+2y-6z=4 x-3y+9z=9 -x-3y+7z=-1

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