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

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

Step 1

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

Add $x$ to both sides of the equation.

$3 y + z = - 3 + x$

$x - 2 y - 2 z = 8$

$3 x - 4 y - z = 6$

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

$z = - 3 + x - 3 y$

$x - 2 y - 2 z = 8$

$3 x - 4 y - z = 6$

$z = - 3 + x - 3 y$

$x - 2 y - 2 z = 8$

$3 x - 4 y - z = 6$

Step 2

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

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

$x - 2 y - 2 (- 3 + x - 3 y) = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

Simplify the left side.

Simplify $x - 2 y - 2 (- 3 + x - 3 y)$ .

Simplify each term.

Apply the distributive property.

$x - 2 y - 2 \cdot - 3 - 2 x - 2 (- 3 y) = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

Simplify.

Multiply $- 2$ by $- 3$ .

$x - 2 y + 6 - 2 x - 2 (- 3 y) = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

Multiply $- 3$ by $- 2$ .

$x - 2 y + 6 - 2 x + 6 y = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

$x - 2 y + 6 - 2 x + 6 y = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

$x - 2 y + 6 - 2 x + 6 y = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

Simplify by adding terms.

Subtract $2 x$ from $x$ .

$- x - 2 y + 6 + 6 y = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

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

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y - z = 6$

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

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

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Simplify the left side.

Simplify $3 x - 4 y - (- 3 + x - 3 y)$ .

Simplify each term.

Apply the distributive property.

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

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Simplify.

Multiply $- 1$ by $- 3$ .

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

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Multiply $- 3$ by $- 1$ .

$3 x - 4 y + 3 - x + 3 y = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y + 3 - x + 3 y = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$3 x - 4 y + 3 - x + 3 y = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Simplify by adding terms.

Subtract $x$ from $3 x$ .

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

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Add $- 4 y$ and $3 y$ .

$2 x - y + 3 = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$2 x - y + 3 = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$2 x - y + 3 = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$2 x - y + 3 = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$2 x - y + 3 = 6$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Step 3

Solve for $x$ in $2 x - y + 3 = 6$ .

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

Add $y$ to both sides of the equation.

$2 x + 3 = 6 + y$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Subtract $3$ from both sides of the equation.

$2 x = 6 + y - 3$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Subtract $3$ from $6$ .

$2 x = y + 3$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$2 x = y + 3$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Divide each term in $2 x = y + 3$ by $2$ and simplify.

Divide each term in $2 x = y + 3$ by $2$ .

$\frac{2 x}{2} = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Divide $x$ by $1$ .

$x = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$x = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$x = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$x = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

$x = \frac{y}{2} + \frac{3}{2}$

$- x + 4 y + 6 = 8$

$z = - 3 + x - 3 y$

Step 4

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

Replace all occurrences of $x$ in $- x + 4 y + 6 = 8$ with $\frac{y}{2} + \frac{3}{2}$ .

$- (\frac{y}{2} + \frac{3}{2}) + 4 y + 6 = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Simplify the left side.

Simplify $- (\frac{y}{2} + \frac{3}{2}) + 4 y + 6$ .

Apply the distributive property.

$- \frac{y}{2} - \frac{3}{2} + 4 y + 6 = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

To write $4 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{y}{2} + 4 y \cdot \frac{2}{2} - \frac{3}{2} + 6 = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Combine $4 y$ and $\frac{2}{2}$ .

$- \frac{y}{2} + \frac{4 y \cdot 2}{2} - \frac{3}{2} + 6 = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Combine the numerators over the common denominator.

$\frac{- y + 4 y \cdot 2}{2} - \frac{3}{2} + 6 = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Combine the numerators over the common denominator.

$6 + \frac{- y + 4 y \cdot 2 - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Multiply $2$ by $4$ .

$6 + \frac{- y + 8 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Add $- y$ and $8 y$ .

$6 + \frac{7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

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

$6 \cdot \frac{2}{2} + \frac{7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Simplify terms.

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

$\frac{6 \cdot 2}{2} + \frac{7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Combine the numerators over the common denominator.

$\frac{6 \cdot 2 + 7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

$\frac{6 \cdot 2 + 7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Simplify the numerator.

Multiply $6$ by $2$ .

$\frac{12 + 7 y - 3}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

Subtract $3$ from $12$ .

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 3 + x - 3 y$

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

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

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify $z = - 3 + \frac{y}{2} + \frac{3}{2} - 3 y$ .

Simplify the left side.

Remove parentheses.

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

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

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

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the right side.

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

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z = \frac{y}{2} - 3 \cdot \frac{2}{2} + \frac{3}{2} - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$z = \frac{y}{2} + \frac{- 3 \cdot 2}{2} + \frac{3}{2} - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$z = \frac{y}{2} + \frac{- 3 \cdot 2 + 3}{2} - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$z = \frac{y}{2} + \frac{- 6 + 3}{2} - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Add $- 6$ and $3$ .

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

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

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

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Move the negative in front of the fraction.

$z = \frac{y}{2} - \frac{3}{2} - 3 y$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z = \frac{y}{2} - 3 y \cdot \frac{2}{2} - \frac{3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Combine $- 3 y$ and $\frac{2}{2}$ .

$z = \frac{y}{2} + \frac{- 3 y \cdot 2}{2} - \frac{3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$z = \frac{y - 3 y \cdot 2}{2} - \frac{3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Combine the numerators over the common denominator.

$z = \frac{y - 3 y \cdot 2 - 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Multiply $2$ by $- 3$ .

$z = \frac{y - 6 y - 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Subtract $6 y$ from $y$ .

$z = \frac{- 5 y - 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Factor $- 1$ out of $- 5 y$ .

$z = \frac{- (5 y) - 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Rewrite $- 3$ as $- 1 (3)$ .

$z = \frac{- (5 y) - 1 \cdot 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Factor $- 1$ out of $- (5 y) - 1 (3)$ .

$z = \frac{- (5 y + 3)}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the expression.

Rewrite $- (5 y + 3)$ as $- 1 (5 y + 3)$ .

$z = \frac{- 1 (5 y + 3)}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Move the negative in front of the fraction.

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{5 y + 3}{2}$

$\frac{7 y + 9}{2} = 8$

$x = \frac{y}{2} + \frac{3}{2}$

Step 5

Solve for $y$ in $\frac{7 y + 9}{2} = 8$ .

Multiply both sides by $2$ .

$\frac{7 y + 9}{2} \cdot 2 = 8 \cdot 2$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{7 y + 9}{2} \cdot 2 = 8 \cdot 2$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Rewrite the expression.

$7 y + 9 = 8 \cdot 2$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$7 y + 9 = 8 \cdot 2$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$7 y + 9 = 8 \cdot 2$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the right side.

Multiply $8$ by $2$ .

$7 y + 9 = 16$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$7 y + 9 = 16$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$7 y + 9 = 16$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Solve for $y$ .

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

Subtract $9$ from both sides of the equation.

$7 y = 16 - 9$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Subtract $9$ from $16$ .

$7 y = 7$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$7 y = 7$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Divide each term in $7 y = 7$ by $7$ and simplify.

Divide each term in $7 y = 7$ by $7$ .

$\frac{7 y}{7} = \frac{7}{7}$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{7}{7}$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Divide $y$ by $1$ .

$y = \frac{7}{7}$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = \frac{7}{7}$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = \frac{7}{7}$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the right side.

Divide $7$ by $7$ .

$y = 1$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = 1$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = 1$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = 1$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

$y = 1$

$z = - \frac{5 y + 3}{2}$

$x = \frac{y}{2} + \frac{3}{2}$

Step 6

Replace all occurrences of $y$ with $1$ in each equation.

Replace all occurrences of $y$ in $z = - \frac{5 y + 3}{2}$ with $1$ .

$z = - \frac{5 (1) + 3}{2}$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the right side.

Simplify $- \frac{5 (1) + 3}{2}$ .

Simplify the numerator.

Multiply $5$ by $1$ .

$z = - \frac{5 + 3}{2}$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

Add $5$ and $3$ .

$z = - \frac{8}{2}$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - \frac{8}{2}$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

Simplify the expression.

Divide $8$ by $2$ .

$z = - 1 \cdot 4$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

Multiply $- 1$ by $4$ .

$z = - 4$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 4$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 4$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

$z = - 4$

$y = 1$

$x = \frac{y}{2} + \frac{3}{2}$

Replace all occurrences of $y$ in $x = \frac{y}{2} + \frac{3}{2}$ with $1$ .

$x = \frac{1}{2} + \frac{3}{2}$

$z = - 4$

$y = 1$

Simplify the right side.

Simplify $\frac{1}{2} + \frac{3}{2}$ .

Combine the numerators over the common denominator.

$x = \frac{1 + 3}{2}$

$z = - 4$

$y = 1$

Simplify the expression.

Add $1$ and $3$ .

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

$z = - 4$

$y = 1$

Divide $4$ by $2$ .

$x = 2$

$z = - 4$

$y = 1$

$x = 2$

$z = - 4$

$y = 1$

$x = 2$

$z = - 4$

$y = 1$

$x = 2$

$z = - 4$

$y = 1$

$x = 2$

$z = - 4$

$y = 1$

Step 7

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

$(2, 1, - 4)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, 1, - 4)$

Equation Form:

$x = 2, y = 1, z = - 4$

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

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