Solve the System of Equations 2x-y+10z=3 x+y+6z=-2 x-y+2z=0

$2 x - y + 10 z = 3$ $x + y + 6 z = - 2$ $x - y + 2 z = 0$

Step 1

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

Subtract $y$ from both sides of the equation.

$x + 6 z = - 2 - y$

$2 x - y + 10 z = 3$

$x - y + 2 z = 0$

Subtract $6 z$ from both sides of the equation.

$x = - 2 - y - 6 z$

$2 x - y + 10 z = 3$

$x - y + 2 z = 0$

$x = - 2 - y - 6 z$

$2 x - y + 10 z = 3$

$x - y + 2 z = 0$

Step 2

Replace all occurrences of $x$ with $- 2 - y - 6 z$ in each equation.

Replace all occurrences of $x$ in $2 x - y + 10 z = 3$ with $- 2 - y - 6 z$ .

$2 (- 2 - y - 6 z) - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Simplify the left side.

Simplify $2 (- 2 - y - 6 z) - y + 10 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot - 2 + 2 (- y) + 2 (- 6 z) - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Simplify.

Multiply $2$ by $- 2$ .

$- 4 + 2 (- y) + 2 (- 6 z) - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Multiply $- 1$ by $2$ .

$- 4 - 2 y + 2 (- 6 z) - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Multiply $- 6$ by $2$ .

$- 4 - 2 y - 12 z - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

$- 4 - 2 y - 12 z - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

$- 4 - 2 y - 12 z - y + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Simplify by adding terms.

Subtract $y$ from $- 2 y$ .

$- 4 - 3 y - 12 z + 10 z = 3$

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Add $- 12 z$ and $10 z$ .

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

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

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

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

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

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

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

$x = - 2 - y - 6 z$

$x - y + 2 z = 0$

Replace all occurrences of $x$ in $x - y + 2 z = 0$ with $- 2 - y - 6 z$ .

$(- 2 - y - 6 z) - y + 2 z = 0$

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

$x = - 2 - y - 6 z$

Simplify the left side.

Simplify $(- 2 - y - 6 z) - y + 2 z$ .

Subtract $y$ from $- y$ .

$- 2 - 2 y - 6 z + 2 z = 0$

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

$x = - 2 - y - 6 z$

Add $- 6 z$ and $2 z$ .

$- 2 - 2 y - 4 z = 0$

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

$x = - 2 - y - 6 z$

$- 2 - 2 y - 4 z = 0$

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

$x = - 2 - y - 6 z$

$- 2 - 2 y - 4 z = 0$

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

$x = - 2 - y - 6 z$

$- 2 - 2 y - 4 z = 0$

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

$x = - 2 - y - 6 z$

Step 3

Solve for $y$ in $- 2 - 2 y - 4 z = 0$ .

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

Add $2$ to both sides of the equation.

$- 2 y - 4 z = 2$

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

$x = - 2 - y - 6 z$

Add $4 z$ to both sides of the equation.

$- 2 y = 2 + 4 z$

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

$x = - 2 - y - 6 z$

$- 2 y = 2 + 4 z$

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

$x = - 2 - y - 6 z$

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

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

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

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

$x = - 2 - y - 6 z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

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

$x = - 2 - y - 6 z$

Divide $y$ by $1$ .

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

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

$x = - 2 - y - 6 z$

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

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

$x = - 2 - y - 6 z$

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

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

$x = - 2 - y - 6 z$

Simplify the right side.

Simplify each term.

Divide $2$ by $- 2$ .

$y = - 1 + \frac{4 z}{- 2}$

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

$x = - 2 - y - 6 z$

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4 z$ .

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

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

$x = - 2 - y - 6 z$

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

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

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

$x = - 2 - y - 6 z$

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

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

$x = - 2 - y - 6 z$

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

$y = - 1 - (2 z)$

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

$x = - 2 - y - 6 z$

Multiply $2$ by $- 1$ .

$y = - 1 - 2 z$

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

$x = - 2 - y - 6 z$

$y = - 1 - 2 z$

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

$x = - 2 - y - 6 z$

$y = - 1 - 2 z$

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

$x = - 2 - y - 6 z$

$y = - 1 - 2 z$

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

$x = - 2 - y - 6 z$

$y = - 1 - 2 z$

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

$x = - 2 - y - 6 z$

Step 4

Replace all occurrences of $y$ with $- 1 - 2 z$ in each equation.

Replace all occurrences of $y$ in $- 4 - 3 y - 2 z = 3$ with $- 1 - 2 z$ .

$- 4 - 3 (- 1 - 2 z) - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Simplify the left side.

Simplify $- 4 - 3 (- 1 - 2 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$- 4 - 3 \cdot - 1 - 3 (- 2 z) - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Multiply $- 3$ by $- 1$ .

$- 4 + 3 - 3 (- 2 z) - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Multiply $- 2$ by $- 3$ .

$- 4 + 3 + 6 z - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

$- 4 + 3 + 6 z - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Simplify by adding terms.

Add $- 4$ and $3$ .

$- 1 + 6 z - 2 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Subtract $2 z$ from $6 z$ .

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 2 - y - 6 z$

Replace all occurrences of $y$ in $x = - 2 - y - 6 z$ with $- 1 - 2 z$ .

$x = - 2 - (- 1 - 2 z) - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Simplify the right side.

Simplify $- 2 - (- 1 - 2 z) - 6 z$ .

Simplify each term.

Apply the distributive property.

$x = - 2 + 1 - (- 2 z) - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Multiply $- 1$ by $- 1$ .

$x = - 2 + 1 - (- 2 z) - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Multiply $- 2$ by $- 1$ .

$x = - 2 + 1 + 2 z - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 2 + 1 + 2 z - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Simplify by adding terms.

Add $- 2$ and $1$ .

$x = - 1 + 2 z - 6 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Subtract $6 z$ from $2 z$ .

$x = - 1 - 4 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 1 - 4 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 1 - 4 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 1 - 4 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

$x = - 1 - 4 z$

$- 1 + 4 z = 3$

$y = - 1 - 2 z$

Step 5

Solve for $z$ in $- 1 + 4 z = 3$ .

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

Add $1$ to both sides of the equation.

$4 z = 3 + 1$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Add $3$ and $1$ .

$4 z = 4$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$4 z = 4$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Divide each term in $4 z = 4$ by $4$ and simplify.

Divide each term in $4 z = 4$ by $4$ .

$\frac{4 z}{4} = \frac{4}{4}$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 z}{4} = \frac{4}{4}$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Divide $z$ by $1$ .

$z = \frac{4}{4}$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$z = \frac{4}{4}$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$z = \frac{4}{4}$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Simplify the right side.

Divide $4$ by $4$ .

$z = 1$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$z = 1$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$z = 1$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

$z = 1$

$x = - 1 - 4 z$

$y = - 1 - 2 z$

Step 6

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

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

$x = - 1 - 4 \cdot 1$

$z = 1$

$y = - 1 - 2 z$

Simplify the right side.

Simplify $- 1 - 4 \cdot 1$ .

Multiply $- 4$ by $1$ .

$x = - 1 - 4$

$z = 1$

$y = - 1 - 2 z$

Subtract $4$ from $- 1$ .

$x = - 5$

$z = 1$

$y = - 1 - 2 z$

$x = - 5$

$z = 1$

$y = - 1 - 2 z$

$x = - 5$

$z = 1$

$y = - 1 - 2 z$

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

$y = - 1 - 2 \cdot 1$

$x = - 5$

$z = 1$

Simplify the right side.

Simplify $- 1 - 2 \cdot 1$ .

Multiply $- 2$ by $1$ .

$y = - 1 - 2$

$x = - 5$

$z = 1$

Subtract $2$ from $- 1$ .

$y = - 3$

$x = - 5$

$z = 1$

$y = - 3$

$x = - 5$

$z = 1$

$y = - 3$

$x = - 5$

$z = 1$

$y = - 3$

$x = - 5$

$z = 1$

Step 7

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

$(- 5, - 3, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 5, - 3, 1)$

Equation Form:

$x = - 5, y = - 3, z = 1$

Solve the System of Equations 2x-y+10z=3 x+y+6z=-2 x-y+2z=0

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