Solve the System of Equations 2x+2y+z=-2 -2x-y+2z=-5 4x+2y+z=0

$2 x + 2 y + z = - 2$ $- 2 x - y + 2 z = - 5$ $4 x + 2 y + z = 0$

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

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

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

$- 2 x - y + 2 z = - 5$

$4 x + 2 y + z = 0$

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

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

$- 2 x - y + 2 z = - 5$

$4 x + 2 y + z = 0$

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

$- 2 x - y + 2 z = - 5$

$4 x + 2 y + z = 0$

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

Replace all occurrences of $z$ in $- 2 x - y + 2 z = - 5$ with $- 2 - 2 x - 2 y$ .

$- 2 x - y + 2 (- 2 - 2 x - 2 y) = - 5$

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

$4 x + 2 y + z = 0$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$- 2 x - y + 2 \cdot - 2 + 2 (- 2 x) + 2 (- 2 y) = - 5$

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

$4 x + 2 y + z = 0$

Simplify.

Multiply $2$ by $- 2$ .

$- 2 x - y - 4 + 2 (- 2 x) + 2 (- 2 y) = - 5$

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

$4 x + 2 y + z = 0$

Multiply $- 2$ by $2$ .

$- 2 x - y - 4 - 4 x + 2 (- 2 y) = - 5$

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

$4 x + 2 y + z = 0$

Multiply $- 2$ by $2$ .

$- 2 x - y - 4 - 4 x - 4 y = - 5$

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

$4 x + 2 y + z = 0$

$- 2 x - y - 4 - 4 x - 4 y = - 5$

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

$4 x + 2 y + z = 0$

$- 2 x - y - 4 - 4 x - 4 y = - 5$

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

$4 x + 2 y + z = 0$

Simplify by adding terms.

Subtract $4 x$ from $- 2 x$ .

$- 6 x - y - 4 - 4 y = - 5$

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

$4 x + 2 y + z = 0$

Subtract $4 y$ from $- y$ .

$- 6 x - 5 y - 4 = - 5$

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

$4 x + 2 y + z = 0$

$- 6 x - 5 y - 4 = - 5$

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

$4 x + 2 y + z = 0$

$- 6 x - 5 y - 4 = - 5$

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

$4 x + 2 y + z = 0$

$- 6 x - 5 y - 4 = - 5$

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

$4 x + 2 y + z = 0$

Replace all occurrences of $z$ in $4 x + 2 y + z = 0$ with $- 2 - 2 x - 2 y$ .

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

$- 6 x - 5 y - 4 = - 5$

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

Simplify the left side.

Simplify $4 x + 2 y - 2 - 2 x - 2 y$ .

Remove parentheses.

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

$- 6 x - 5 y - 4 = - 5$

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

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

Subtract $2 y$ from $2 y$ .

$4 x - 2 - 2 x + 0 = 0$

$- 6 x - 5 y - 4 = - 5$

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

Add $4 x - 2 - 2 x$ and $0$ .

$4 x - 2 - 2 x = 0$

$- 6 x - 5 y - 4 = - 5$

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

$4 x - 2 - 2 x = 0$

$- 6 x - 5 y - 4 = - 5$

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

Subtract $2 x$ from $4 x$ .

$2 x - 2 = 0$

$- 6 x - 5 y - 4 = - 5$

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

$2 x - 2 = 0$

$- 6 x - 5 y - 4 = - 5$

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

$2 x - 2 = 0$

$- 6 x - 5 y - 4 = - 5$

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

$2 x - 2 = 0$

$- 6 x - 5 y - 4 = - 5$

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

Solve for $x$ in $2 x - 2 = 0$ .

Add $2$ to both sides of the equation.

$2 x = 2$

$- 6 x - 5 y - 4 = - 5$

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

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

Divide each term in $2 x = 2$ by $2$ .

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

$- 6 x - 5 y - 4 = - 5$

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$- 6 x - 5 y - 4 = - 5$

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

Divide $x$ by $1$ .

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

$- 6 x - 5 y - 4 = - 5$

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

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

$- 6 x - 5 y - 4 = - 5$

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

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

$- 6 x - 5 y - 4 = - 5$

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

Simplify the right side.

Divide $2$ by $2$ .

$x = 1$

$- 6 x - 5 y - 4 = - 5$

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

$x = 1$

$- 6 x - 5 y - 4 = - 5$

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

$x = 1$

$- 6 x - 5 y - 4 = - 5$

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

$x = 1$

$- 6 x - 5 y - 4 = - 5$

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

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

Replace all occurrences of $x$ in $- 6 x - 5 y - 4 = - 5$ with $1$ .

$- 6 \cdot 1 - 5 y - 4 = - 5$

$x = 1$

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

Simplify the left side.

Simplify $- 6 \cdot 1 - 5 y - 4$ .

Multiply $- 6$ by $1$ .

$- 6 - 5 y - 4 = - 5$

$x = 1$

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

Subtract $4$ from $- 6$ .

$- 5 y - 10 = - 5$

$x = 1$

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

$- 5 y - 10 = - 5$

$x = 1$

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

$- 5 y - 10 = - 5$

$x = 1$

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

Replace all occurrences of $x$ in $z = - 2 - 2 x - 2 y$ with $1$ .

$z = - 2 - 2 \cdot 1 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

Simplify the right side.

Simplify $- 2 - 2 \cdot 1 - 2 y$ .

Multiply $- 2$ by $1$ .

$z = - 2 - 2 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

Subtract $2$ from $- 2$ .

$z = - 4 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

$z = - 4 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

$z = - 4 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

$z = - 4 - 2 y$

$- 5 y - 10 = - 5$

$x = 1$

Solve for $y$ in $- 5 y - 10 = - 5$ .

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

Add $10$ to both sides of the equation.

$- 5 y = - 5 + 10$

$z = - 4 - 2 y$

$x = 1$

Add $- 5$ and $10$ .

$- 5 y = 5$

$z = - 4 - 2 y$

$x = 1$

$- 5 y = 5$

$z = - 4 - 2 y$

$x = 1$

Divide each term in $- 5 y = 5$ by $- 5$ and simplify.

Divide each term in $- 5 y = 5$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{5}{- 5}$

$z = - 4 - 2 y$

$x = 1$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{5}{- 5}$

$z = - 4 - 2 y$

$x = 1$

Divide $y$ by $1$ .

$y = \frac{5}{- 5}$

$z = - 4 - 2 y$

$x = 1$

$y = \frac{5}{- 5}$

$z = - 4 - 2 y$

$x = 1$

$y = \frac{5}{- 5}$

$z = - 4 - 2 y$

$x = 1$

Simplify the right side.

Divide $5$ by $- 5$ .

$y = - 1$

$z = - 4 - 2 y$

$x = 1$

$y = - 1$

$z = - 4 - 2 y$

$x = 1$

$y = - 1$

$z = - 4 - 2 y$

$x = 1$

$y = - 1$

$z = - 4 - 2 y$

$x = 1$

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

Replace all occurrences of $y$ in $z = - 4 - 2 y$ with $- 1$ .

$z = - 4 - 2 \cdot - 1$

$y = - 1$

$x = 1$

Simplify the right side.

Simplify $- 4 - 2 \cdot - 1$ .

Multiply $- 2$ by $- 1$ .

$z = - 4 + 2$

$y = - 1$

$x = 1$

Add $- 4$ and $2$ .

$z = - 2$

$y = - 1$

$x = 1$

$z = - 2$

$y = - 1$

$x = 1$

$z = - 2$

$y = - 1$

$x = 1$

$z = - 2$

$y = - 1$

$x = 1$

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

$(1, - 1, - 2)$

The result can be shown in multiple forms.

Point Form:

$(1, - 1, - 2)$

Equation Form:

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

Solve the System of Equations 2x+2y+z=-2 -2x-y+2z=-5 4x+2y+z=0

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