Solve the System of Equations 2x-4y+2z=16 -2x+5y+2z=-34 x-2y+2z=4

$2 x - 4 y + 2 z = 16$ $- 2 x + 5 y + 2 z = - 34$ $x - 2 y + 2 z = 4$

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

Add $2 y$ to both sides of the equation.

$x + 2 z = 4 + 2 y$

$2 x - 4 y + 2 z = 16$

$- 2 x + 5 y + 2 z = - 34$

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

$x = 4 + 2 y - 2 z$

$2 x - 4 y + 2 z = 16$

$- 2 x + 5 y + 2 z = - 34$

$x = 4 + 2 y - 2 z$

$2 x - 4 y + 2 z = 16$

$- 2 x + 5 y + 2 z = - 34$

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

Replace all occurrences of $x$ in $2 x - 4 y + 2 z = 16$ with $4 + 2 y - 2 z$ .

$2 (4 + 2 y - 2 z) - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Simplify the left side.

Simplify $2 (4 + 2 y - 2 z) - 4 y + 2 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 4 + 2 (2 y) + 2 (- 2 z) - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Simplify.

Multiply $2$ by $4$ .

$8 + 2 (2 y) + 2 (- 2 z) - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Multiply $2$ by $2$ .

$8 + 4 y + 2 (- 2 z) - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Multiply $- 2$ by $2$ .

$8 + 4 y - 4 z - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 + 4 y - 4 z - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 + 4 y - 4 z - 4 y + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Simplify by adding terms.

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

Subtract $4 y$ from $4 y$ .

$8 - 4 z + 0 + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Add $8 - 4 z$ and $0$ .

$8 - 4 z + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 - 4 z + 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Add $- 4 z$ and $2 z$ .

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 2 x + 5 y + 2 z = - 34$

Replace all occurrences of $x$ in $- 2 x + 5 y + 2 z = - 34$ with $4 + 2 y - 2 z$ .

$- 2 (4 + 2 y - 2 z) + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Simplify the left side.

Simplify $- 2 (4 + 2 y - 2 z) + 5 y + 2 z$ .

Simplify each term.

Apply the distributive property.

$- 2 \cdot 4 - 2 (2 y) - 2 (- 2 z) + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Simplify.

Multiply $- 2$ by $4$ .

$- 8 - 2 (2 y) - 2 (- 2 z) + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Multiply $2$ by $- 2$ .

$- 8 - 4 y - 2 (- 2 z) + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Multiply $- 2$ by $- 2$ .

$- 8 - 4 y + 4 z + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 - 4 y + 4 z + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 - 4 y + 4 z + 5 y + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Simplify by adding terms.

Add $- 4 y$ and $5 y$ .

$- 8 + y + 4 z + 2 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Add $4 z$ and $2 z$ .

$- 8 + y + 6 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 + y + 6 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 + y + 6 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 + y + 6 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

$- 8 + y + 6 z = - 34$

$8 - 2 z = 16$

$x = 4 + 2 y - 2 z$

Solve for $z$ in $8 - 2 z = 16$ .

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

Subtract $8$ from both sides of the equation.

$- 2 z = 16 - 8$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

Subtract $8$ from $16$ .

$- 2 z = 8$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

$- 2 z = 8$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

Divide each term in $- 2 z = 8$ by $- 2$ and simplify.

Divide each term in $- 2 z = 8$ by $- 2$ .

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

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

Divide $z$ by $1$ .

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

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

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

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

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

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

Simplify the right side.

Divide $8$ by $- 2$ .

$z = - 4$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

$z = - 4$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

$z = - 4$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

$z = - 4$

$- 8 + y + 6 z = - 34$

$x = 4 + 2 y - 2 z$

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

Replace all occurrences of $z$ in $- 8 + y + 6 z = - 34$ with $- 4$ .

$- 8 + y + 6 (- 4) = - 34$

$z = - 4$

$x = 4 + 2 y - 2 z$

Simplify the left side.

Simplify $- 8 + y + 6 (- 4)$ .

Multiply $6$ by $- 4$ .

$- 8 + y - 24 = - 34$

$z = - 4$

$x = 4 + 2 y - 2 z$

Subtract $24$ from $- 8$ .

$y - 32 = - 34$

$z = - 4$

$x = 4 + 2 y - 2 z$

$y - 32 = - 34$

$z = - 4$

$x = 4 + 2 y - 2 z$

$y - 32 = - 34$

$z = - 4$

$x = 4 + 2 y - 2 z$

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

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

$y - 32 = - 34$

$z = - 4$

Simplify the right side.

Simplify $4 + 2 y - 2 \cdot - 4$ .

Multiply $- 2$ by $- 4$ .

$x = 4 + 2 y + 8$

$y - 32 = - 34$

$z = - 4$

Add $4$ and $8$ .

$x = 2 y + 12$

$y - 32 = - 34$

$z = - 4$

$x = 2 y + 12$

$y - 32 = - 34$

$z = - 4$

$x = 2 y + 12$

$y - 32 = - 34$

$z = - 4$

$x = 2 y + 12$

$y - 32 = - 34$

$z = - 4$

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

Add $32$ to both sides of the equation.

$y = - 34 + 32$

$x = 2 y + 12$

$z = - 4$

Add $- 34$ and $32$ .

$y = - 2$

$x = 2 y + 12$

$z = - 4$

$y = - 2$

$x = 2 y + 12$

$z = - 4$

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

Replace all occurrences of $y$ in $x = 2 y + 12$ with $- 2$ .

$x = 2 (- 2) + 12$

$y = - 2$

$z = - 4$

Simplify the right side.

Simplify $2 (- 2) + 12$ .

Multiply $2$ by $- 2$ .

$x = - 4 + 12$

$y = - 2$

$z = - 4$

Add $- 4$ and $12$ .

$x = 8$

$y = - 2$

$z = - 4$

$x = 8$

$y = - 2$

$z = - 4$

$x = 8$

$y = - 2$

$z = - 4$

$x = 8$

$y = - 2$

$z = - 4$

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

$(8, - 2, - 4)$

The result can be shown in multiple forms.

Point Form:

$(8, - 2, - 4)$

Equation Form:

$x = 8, y = - 2, z = - 4$

Solve the System of Equations 2x-4y+2z=16 -2x+5y+2z=-34 x-2y+2z=4

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