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

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

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

Add $4 x$ to both sides of the equation.

$y + 3 z = 8 + 4 x$

$3 x + 3 y - 6 z = 9$

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

Subtract $3 z$ from both sides of the equation.

$y = 8 + 4 x - 3 z$

$3 x + 3 y - 6 z = 9$

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

$y = 8 + 4 x - 3 z$

$3 x + 3 y - 6 z = 9$

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

Replace all occurrences of $y$ with $8 + 4 x - 3 z$ in each equation.

Replace all occurrences of $y$ in $3 x + 3 y - 6 z = 9$ with $8 + 4 x - 3 z$ .

$3 x + 3 (8 + 4 x - 3 z) - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Simplify the left side.

Simplify $3 x + 3 (8 + 4 x - 3 z) - 6 z$ .

Simplify each term.

Apply the distributive property.

$3 x + 3 \cdot 8 + 3 (4 x) + 3 (- 3 z) - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Simplify.

Multiply $3$ by $8$ .

$3 x + 24 + 3 (4 x) + 3 (- 3 z) - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Multiply $4$ by $3$ .

$3 x + 24 + 12 x + 3 (- 3 z) - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Multiply $- 3$ by $3$ .

$3 x + 24 + 12 x - 9 z - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

$3 x + 24 + 12 x - 9 z - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

$3 x + 24 + 12 x - 9 z - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Simplify by adding terms.

Add $3 x$ and $12 x$ .

$15 x + 24 - 9 z - 6 z = 9$

$y = 8 + 4 x - 3 z$

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

Subtract $6 z$ from $- 9 z$ .

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

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

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

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

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

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

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

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

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

$- 2 x + 8 + 4 x - 3 z + 2 z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Simplify the left side.

Simplify $- 2 x + 8 + 4 x - 3 z + 2 z$ .

Remove parentheses.

$- 2 x + 8 + 4 x - 3 z + 2 z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Add $- 2 x$ and $4 x$ .

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

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Add $- 3 z$ and $2 z$ .

$2 x + 8 - z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$2 x + 8 - z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$2 x + 8 - z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$2 x + 8 - z = 2$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Solve for $x$ in $2 x + 8 - z = 2$ .

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

Subtract $8$ from both sides of the equation.

$2 x - z = 2 - 8$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Add $z$ to both sides of the equation.

$2 x = 2 - 8 + z$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Subtract $8$ from $2$ .

$2 x = - 6 + z$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$2 x = - 6 + z$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Divide each term in $2 x = - 6 + z$ by $2$ and simplify.

Divide each term in $2 x = - 6 + z$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2} + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2} + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Divide $x$ by $1$ .

$x = \frac{- 6}{2} + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$x = \frac{- 6}{2} + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$x = \frac{- 6}{2} + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Simplify the right side.

Divide $- 6$ by $2$ .

$x = - 3 + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$x = - 3 + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$x = - 3 + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

$x = - 3 + \frac{z}{2}$

$15 x + 24 - 15 z = 9$

$y = 8 + 4 x - 3 z$

Replace all occurrences of $x$ with $- 3 + \frac{z}{2}$ in each equation.

Replace all occurrences of $x$ in $15 x + 24 - 15 z = 9$ with $- 3 + \frac{z}{2}$ .

$15 (- 3 + \frac{z}{2}) + 24 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Simplify the left side.

Simplify $15 (- 3 + \frac{z}{2}) + 24 - 15 z$ .

Simplify each term.

Apply the distributive property.

$15 \cdot - 3 + 15 (\frac{z}{2}) + 24 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Multiply $15$ by $- 3$ .

$- 45 + 15 (\frac{z}{2}) + 24 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Combine $15$ and $\frac{z}{2}$ .

$- 45 + \frac{15 z}{2} + 24 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$- 45 + \frac{15 z}{2} + 24 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Add $- 45$ and $24$ .

$\frac{15 z}{2} - 21 - 15 z = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

To write $- 15 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15 z}{2} - 15 z \cdot \frac{2}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Simplify terms.

Combine $- 15 z$ and $\frac{2}{2}$ .

$\frac{15 z}{2} + \frac{- 15 z \cdot 2}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Combine the numerators over the common denominator.

$\frac{15 z - 15 z \cdot 2}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$\frac{15 z - 15 z \cdot 2}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Simplify each term.

Simplify the numerator.

Factor $15 z$ out of $15 z - 15 z \cdot 2$ .

Factor $15 z$ out of $15 z$ .

$\frac{15 z (1) - 15 z \cdot 2}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Factor $15 z$ out of $- 15 z \cdot 2$ .

$\frac{15 z (1) + 15 z (- 1 \cdot 2)}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Factor $15 z$ out of $15 z (1) + 15 z (- 1 \cdot 2)$ .

$\frac{15 z (1 - 1 \cdot 2)}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$\frac{15 z (1 - 1 \cdot 2)}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Multiply $- 1$ by $2$ .

$\frac{15 z (1 - 2)}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Subtract $2$ from $1$ .

$\frac{15 z \cdot - 1}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Multiply $- 1$ by $15$ .

$\frac{- 15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$\frac{- 15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Move the negative in front of the fraction.

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 + 4 x - 3 z$

Replace all occurrences of $x$ in $y = 8 + 4 x - 3 z$ with $- 3 + \frac{z}{2}$ .

$y = 8 + 4 (- 3 + \frac{z}{2}) - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Simplify the right side.

Simplify $8 + 4 (- 3 + \frac{z}{2}) - 3 z$ .

Simplify each term.

Apply the distributive property.

$y = 8 + 4 \cdot - 3 + 4 (\frac{z}{2}) - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Multiply $4$ by $- 3$ .

$y = 8 - 12 + 4 (\frac{z}{2}) - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$y = 8 - 12 + 2 (2) (\frac{z}{2}) - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Cancel the common factor.

$y = 8 - 12 + 2 \cdot (2 (\frac{z}{2})) - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Rewrite the expression.

$y = 8 - 12 + 2 z - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 - 12 + 2 z - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = 8 - 12 + 2 z - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Simplify by adding terms.

Subtract $12$ from $8$ .

$y = - 4 + 2 z - 3 z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Subtract $3 z$ from $2 z$ .

$y = - 4 - z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = - 4 - z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = - 4 - z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = - 4 - z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

$y = - 4 - z$

$- \frac{15 z}{2} - 21 = 9$

$x = - 3 + \frac{z}{2}$

Solve for $z$ in $- \frac{15 z}{2} - 21 = 9$ .

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

Add $21$ to both sides of the equation.

$- \frac{15 z}{2} = 9 + 21$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Add $9$ and $21$ .

$- \frac{15 z}{2} = 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$- \frac{15 z}{2} = 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Multiply both sides of the equation by $- \frac{2}{15}$ .

$- \frac{2}{15} \cdot (- \frac{15 z}{2}) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{15} (- \frac{15 z}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{15}$ into the numerator.

$\frac{- 2}{15} \cdot (- \frac{15 z}{2}) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Move the leading negative in $- \frac{15 z}{2}$ into the numerator.

$\frac{- 2}{15} \cdot \frac{- 15 z}{2} = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{15} \cdot \frac{- 15 z}{2} = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Cancel the common factor.

$\frac{2 \cdot - 1}{15} \cdot \frac{- 15 z}{2} = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Rewrite the expression.

$\frac{- 1}{15} \cdot (- 15 z) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$\frac{- 1}{15} \cdot (- 15 z) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Cancel the common factor of $15$ .

Factor $15$ out of $- 15 z$ .

$\frac{- 1}{15} \cdot (15 (- z)) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Cancel the common factor.

$\frac{- 1}{15} \cdot (15 (- z)) = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Rewrite the expression.

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Multiply $z$ by $1$ .

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - \frac{2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Simplify the right side.

Simplify $- \frac{2}{15} \cdot 30$ .

Cancel the common factor of $15$ .

Move the leading negative in $- \frac{2}{15}$ into the numerator.

$z = \frac{- 2}{15} \cdot 30$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Factor $15$ out of $30$ .

$z = \frac{- 2}{15} \cdot (15 (2))$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Cancel the common factor.

$z = \frac{- 2}{15} \cdot (15 \cdot 2)$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Rewrite the expression.

$z = - 2 \cdot 2$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - 2 \cdot 2$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

Multiply $- 2$ by $2$ .

$z = - 4$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - 4$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - 4$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - 4$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

$z = - 4$

$y = - 4 - z$

$x = - 3 + \frac{z}{2}$

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

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

$y = - 4 - (- 4)$

$z = - 4$

$x = - 3 + \frac{z}{2}$

Simplify the right side.

Simplify $- 4 - (- 4)$ .

Multiply $- 1$ by $- 4$ .

$y = - 4 + 4$

$z = - 4$

$x = - 3 + \frac{z}{2}$

Add $- 4$ and $4$ .

$y = 0$

$z = - 4$

$x = - 3 + \frac{z}{2}$

$y = 0$

$z = - 4$

$x = - 3 + \frac{z}{2}$

$y = 0$

$z = - 4$

$x = - 3 + \frac{z}{2}$

Replace all occurrences of $z$ in $x = - 3 + \frac{z}{2}$ with $- 4$ .

$x = - 3 + \frac{- 4}{2}$

$y = 0$

$z = - 4$

Simplify the right side.

Simplify $- 3 + \frac{- 4}{2}$ .

Divide $- 4$ by $2$ .

$x = - 3 - 2$

$y = 0$

$z = - 4$

Subtract $2$ from $- 3$ .

$x = - 5$

$y = 0$

$z = - 4$

$x = - 5$

$y = 0$

$z = - 4$

$x = - 5$

$y = 0$

$z = - 4$

$x = - 5$

$y = 0$

$z = - 4$

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

$(- 5, 0, - 4)$

The result can be shown in multiple forms.

Point Form:

$(- 5, 0, - 4)$

Equation Form:

$x = - 5, y = 0, z = - 4$

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

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