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

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

Solve for $x$ in $2 x + 2 y - z = 4$ .

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

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

$2 x - z = 4 - 2 y$

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

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

Add $z$ to both sides of the equation.

$2 x = 4 - 2 y + z$

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

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

$2 x = 4 - 2 y + z$

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

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

Divide each term in $2 x = 4 - 2 y + z$ by $2$ and simplify.

Divide each term in $2 x = 4 - 2 y + z$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2} + \frac{- 2 y}{2} + \frac{z}{2}$

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2} + \frac{- 2 y}{2} + \frac{z}{2}$

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

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

Divide $x$ by $1$ .

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{z}{2}$

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

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

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{z}{2}$

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

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

$x = \frac{4}{2} + \frac{- 2 y}{2} + \frac{z}{2}$

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

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

Simplify the right side.

Simplify each term.

Divide $4$ by $2$ .

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

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

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

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 y$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

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

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

Cancel the common factor.

$x = 2 + \frac{2 (- y)}{2 \cdot 1} + \frac{z}{2}$

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

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

Rewrite the expression.

$x = 2 + \frac{- y}{1} + \frac{z}{2}$

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

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

Divide $- y$ by $1$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

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

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

Simplify.

Multiply $4$ by $2$ .

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

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

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

Multiply $- 1$ by $4$ .

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

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

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

Cancel the common factor.

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

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

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

Rewrite the expression.

$8 - 4 y + 2 z - 2 y - 2 z = 2$

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

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

$8 - 4 y + 2 z - 2 y - 2 z = 2$

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

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

$8 - 4 y + 2 z - 2 y - 2 z = 2$

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

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

$8 - 4 y + 2 z - 2 y - 2 z = 2$

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

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

Simplify by adding terms.

Combine the opposite terms in $8 - 4 y + 2 z - 2 y - 2 z$ .

Subtract $2 z$ from $2 z$ .

$8 - 4 y - 2 y + 0 = 2$

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

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

Add $8 - 4 y - 2 y$ and $0$ .

$8 - 4 y - 2 y = 2$

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

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

$8 - 4 y - 2 y = 2$

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

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

Subtract $2 y$ from $- 4 y$ .

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

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

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

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

$8 - 6 y = 2$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$8 - 6 y = 2$

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

Simplify.

Multiply $3$ by $2$ .

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

$8 - 6 y = 2$

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

Multiply $- 1$ by $3$ .

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

$8 - 6 y = 2$

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

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

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

Combine the opposite terms in $6 - 3 y + \frac{3 z}{2} + 3 y - 4 z$ .

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

$6 + \frac{3 z}{2} + 0 - 4 z = - 4$

$8 - 6 y = 2$

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

Add $6 + \frac{3 z}{2}$ and $0$ .

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

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

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

$8 - 6 y = 2$

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

Simplify terms.

Combine $- 4 z$ and $\frac{2}{2}$ .

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

$8 - 6 y = 2$

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

Combine the numerators over the common denominator.

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

$8 - 6 y = 2$

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

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

$8 - 6 y = 2$

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

Simplify each term.

Simplify the numerator.

Factor $z$ out of $3 z - 4 z \cdot 2$ .

Factor $z$ out of $3 z$ .

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

$8 - 6 y = 2$

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

Factor $z$ out of $- 4 z \cdot 2$ .

$6 + \frac{z \cdot 3 + z (- 4 \cdot 2)}{2} = - 4$

$8 - 6 y = 2$

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

Factor $z$ out of $z \cdot 3 + z (- 4 \cdot 2)$ .

$6 + \frac{z (3 - 4 \cdot 2)}{2} = - 4$

$8 - 6 y = 2$

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

$6 + \frac{z (3 - 4 \cdot 2)}{2} = - 4$

$8 - 6 y = 2$

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

Multiply $- 4$ by $2$ .

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

$8 - 6 y = 2$

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

Subtract $8$ from $3$ .

$6 + \frac{z \cdot - 5}{2} = - 4$

$8 - 6 y = 2$

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

$6 + \frac{z \cdot - 5}{2} = - 4$

$8 - 6 y = 2$

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

Move $- 5$ to the left of $z$ .

$6 + \frac{- 5 \cdot z}{2} = - 4$

$8 - 6 y = 2$

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

Move the negative in front of the fraction.

$6 - \frac{5 z}{2} = - 4$

$8 - 6 y = 2$

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

$6 - \frac{5 z}{2} = - 4$

$8 - 6 y = 2$

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

$6 - \frac{5 z}{2} = - 4$

$8 - 6 y = 2$

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

$6 - \frac{5 z}{2} = - 4$

$8 - 6 y = 2$

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

$6 - \frac{5 z}{2} = - 4$

$8 - 6 y = 2$

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

Solve for $z$ in $6 - \frac{5 z}{2} = - 4$ .

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

Subtract $6$ from both sides of the equation.

$- \frac{5 z}{2} = - 4 - 6$

$8 - 6 y = 2$

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

Subtract $6$ from $- 4$ .

$- \frac{5 z}{2} = - 10$

$8 - 6 y = 2$

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

$- \frac{5 z}{2} = - 10$

$8 - 6 y = 2$

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

Multiply both sides of the equation by $- \frac{2}{5}$ .

$- \frac{2}{5} \cdot (- \frac{5 z}{2}) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{5} (- \frac{5 z}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$\frac{- 2}{5} \cdot (- \frac{5 z}{2}) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Move the leading negative in $- \frac{5 z}{2}$ into the numerator.

$\frac{- 2}{5} \cdot \frac{- 5 z}{2} = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{5} \cdot \frac{- 5 z}{2} = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Cancel the common factor.

$\frac{2 \cdot - 1}{5} \cdot \frac{- 5 z}{2} = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Rewrite the expression.

$\frac{- 1}{5} \cdot (- 5 z) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

$\frac{- 1}{5} \cdot (- 5 z) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Cancel the common factor of $5$ .

Factor $5$ out of $- 5 z$ .

$\frac{- 1}{5} \cdot (5 (- z)) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Cancel the common factor.

$\frac{- 1}{5} \cdot (5 (- z)) = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Rewrite the expression.

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Multiply $z$ by $1$ .

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

$z = - \frac{2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Simplify the right side.

Simplify $- \frac{2}{5} \cdot - 10$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$z = \frac{- 2}{5} \cdot - 10$

$8 - 6 y = 2$

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

Factor $5$ out of $- 10$ .

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

$8 - 6 y = 2$

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

Cancel the common factor.

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

$8 - 6 y = 2$

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

Rewrite the expression.

$z = - 2 \cdot - 2$

$8 - 6 y = 2$

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

$z = - 2 \cdot - 2$

$8 - 6 y = 2$

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

Multiply $- 2$ by $- 2$ .

$z = 4$

$8 - 6 y = 2$

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

$z = 4$

$8 - 6 y = 2$

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

$z = 4$

$8 - 6 y = 2$

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

$z = 4$

$8 - 6 y = 2$

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

$z = 4$

$8 - 6 y = 2$

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

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

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

Subtract $8$ from both sides of the equation.

$- 6 y = 2 - 8$

$z = 4$

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

Subtract $8$ from $2$ .

$- 6 y = - 6$

$z = 4$

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

$- 6 y = - 6$

$z = 4$

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

Divide each term in $- 6 y = - 6$ by $- 6$ and simplify.

Divide each term in $- 6 y = - 6$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- 6}{- 6}$

$z = 4$

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- 6}{- 6}$

$z = 4$

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

Divide $y$ by $1$ .

$y = \frac{- 6}{- 6}$

$z = 4$

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

$y = \frac{- 6}{- 6}$

$z = 4$

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

$y = \frac{- 6}{- 6}$

$z = 4$

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

Simplify the right side.

Divide $- 6$ by $- 6$ .

$y = 1$

$z = 4$

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

$y = 1$

$z = 4$

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

$y = 1$

$z = 4$

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

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

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

$y = 1$

$z = 4$

Simplify the right side.

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

Divide $4$ by $2$ .

$x = 2 - y + 2$

$y = 1$

$z = 4$

Add $2$ and $2$ .

$x = - y + 4$

$y = 1$

$z = 4$

$x = - y + 4$

$y = 1$

$z = 4$

$x = - y + 4$

$y = 1$

$z = 4$

$x = - y + 4$

$y = 1$

$z = 4$

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

Replace all occurrences of $y$ in $x = - y + 4$ with $1$ .

$x = - (1) + 4$

$y = 1$

$z = 4$

Simplify the right side.

Simplify $- (1) + 4$ .

Multiply $- 1$ by $1$ .

$x = - 1 + 4$

$y = 1$

$z = 4$

Add $- 1$ and $4$ .

$x = 3$

$y = 1$

$z = 4$

$x = 3$

$y = 1$

$z = 4$

$x = 3$

$y = 1$

$z = 4$

$x = 3$

$y = 1$

$z = 4$

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

$(3, 1, 4)$

The result can be shown in multiple forms.

Point Form:

$(3, 1, 4)$

Equation Form:

$x = 3, y = 1, z = 4$

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

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