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

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

Step 1

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

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

$x + 2 z = 4 - 2 y$

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

$x + y + 4 z = 2$

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

$x = 4 - 2 y - 2 z$

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

$x + y + 4 z = 2$

$x = 4 - 2 y - 2 z$

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

$x + y + 4 z = 2$

Step 2

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

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

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

$x = 4 - 2 y - 2 z$

$x + y + 4 z = 2$

Simplify the left side.

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

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

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

$x = 4 - 2 y - 2 z$

$x + y + 4 z = 2$

Subtract $2 z$ from $- 2 z$ .

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

$x = 4 - 2 y - 2 z$

$x + y + 4 z = 2$

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

$x = 4 - 2 y - 2 z$

$x + y + 4 z = 2$

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

$x = 4 - 2 y - 2 z$

$x + y + 4 z = 2$

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

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

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

$x = 4 - 2 y - 2 z$

Simplify the left side.

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

Add $- 2 y$ and $y$ .

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

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

$x = 4 - 2 y - 2 z$

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

$4 - y + 2 z = 2$

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

$x = 4 - 2 y - 2 z$

$4 - y + 2 z = 2$

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

$x = 4 - 2 y - 2 z$

$4 - y + 2 z = 2$

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

$x = 4 - 2 y - 2 z$

$4 - y + 2 z = 2$

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

$x = 4 - 2 y - 2 z$

Step 3

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

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

Subtract $4$ from both sides of the equation.

$- y + 2 z = 2 - 4$

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

$x = 4 - 2 y - 2 z$

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

$- y = 2 - 4 - 2 z$

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

$x = 4 - 2 y - 2 z$

Subtract $4$ from $2$ .

$- y = - 2 - 2 z$

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

$x = 4 - 2 y - 2 z$

$- y = - 2 - 2 z$

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

$x = 4 - 2 y - 2 z$

Divide each term in $- y = - 2 - 2 z$ by $- 1$ and simplify.

Divide each term in $- y = - 2 - 2 z$ by $- 1$ .

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

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

$x = 4 - 2 y - 2 z$

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

$x = 4 - 2 y - 2 z$

Divide $y$ by $1$ .

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

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

$x = 4 - 2 y - 2 z$

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

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

$x = 4 - 2 y - 2 z$

Simplify the right side.

Simplify each term.

Divide $- 2$ by $- 1$ .

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

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

$x = 4 - 2 y - 2 z$

Move the negative one from the denominator of $\frac{- 2 z}{- 1}$ .

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

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

$x = 4 - 2 y - 2 z$

Rewrite $- 1 \cdot (- 2 z)$ as $- (- 2 z)$ .

$y = 2 - (- 2 z)$

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

$x = 4 - 2 y - 2 z$

Multiply $- 2$ by $- 1$ .

$y = 2 + 2 z$

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

$x = 4 - 2 y - 2 z$

$y = 2 + 2 z$

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

$x = 4 - 2 y - 2 z$

$y = 2 + 2 z$

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

$x = 4 - 2 y - 2 z$

$y = 2 + 2 z$

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

$x = 4 - 2 y - 2 z$

$y = 2 + 2 z$

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

$x = 4 - 2 y - 2 z$

Step 4

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

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

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

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Multiply $2$ by $2$ .

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

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Multiply $2$ by $2$ .

$4 + 4 + 4 z - 4 z = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

$4 + 4 + 4 z - 4 z = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Simplify by adding terms.

Combine the opposite terms in $4 + 4 + 4 z - 4 z$ .

Subtract $4 z$ from $4 z$ .

$4 + 4 + 0 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Add $4 + 4$ and $0$ .

$4 + 4 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

$4 + 4 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

Add $4$ and $4$ .

$8 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

$8 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

$8 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

$8 = - 2$

$y = 2 + 2 z$

$x = 4 - 2 y - 2 z$

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

$x = 4 - 2 (2 + 2 z) - 2 z$

$8 = - 2$

$y = 2 + 2 z$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

$x = 4 - 2 \cdot 2 - 2 (2 z) - 2 z$

$8 = - 2$

$y = 2 + 2 z$

Multiply $- 2$ by $2$ .

$x = 4 - 4 - 2 (2 z) - 2 z$

$8 = - 2$

$y = 2 + 2 z$

Multiply $2$ by $- 2$ .

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

$8 = - 2$

$y = 2 + 2 z$

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

$8 = - 2$

$y = 2 + 2 z$

Simplify by adding terms.

Subtract $4$ from $4$ .

$x = 0 - 4 z - 2 z$

$8 = - 2$

$y = 2 + 2 z$

Subtract $4 z$ from $0$ .

$x = - 4 z - 2 z$

$8 = - 2$

$y = 2 + 2 z$

Subtract $2 z$ from $- 4 z$ .

$x = - 6 z$

$8 = - 2$

$y = 2 + 2 z$

$x = - 6 z$

$8 = - 2$

$y = 2 + 2 z$

$x = - 6 z$

$8 = - 2$

$y = 2 + 2 z$

$x = - 6 z$

$8 = - 2$

$y = 2 + 2 z$

$x = - 6 z$

$8 = - 2$

$y = 2 + 2 z$

Step 5

Since $8 = - 2$ is not true, there is no solution.

No solution

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

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