Solve the System of Equations -x+3y+4z=0 x+y-2z=-8 -2x+4y+3z=-4

$- x + 3 y + 4 z = 0$ $x + y - 2 z = - 8$ $- 2 x + 4 y + 3 z = - 4$

Step 1

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

Subtract $y$ from both sides of the equation.

$x - 2 z = - 8 - y$

$- x + 3 y + 4 z = 0$

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

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

$x = - 8 - y + 2 z$

$- x + 3 y + 4 z = 0$

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

$x = - 8 - y + 2 z$

$- x + 3 y + 4 z = 0$

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

Step 2

Replace all occurrences of $x$ with $- 8 - y + 2 z$ in each equation.

Replace all occurrences of $x$ in $- x + 3 y + 4 z = 0$ with $- 8 - y + 2 z$ .

$- (- 8 - y + 2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Simplify the left side.

Simplify $- (- 8 - y + 2 z) + 3 y + 4 z$ .

Simplify each term.

Apply the distributive property.

$8 + y - (2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Simplify.

Multiply $- 1$ by $- 8$ .

$8 + y - (2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$8 + 1 y - (2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Multiply $y$ by $1$ .

$8 + y - (2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

$8 + y - (2 z) + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Multiply $2$ by $- 1$ .

$8 + y - 2 z + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

$8 + y - 2 z + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

$8 + y - 2 z + 3 y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Simplify by adding terms.

Add $y$ and $3 y$ .

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

$x = - 8 - y + 2 z$

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

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

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

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

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

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

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

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

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

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

Replace all occurrences of $x$ in $- 2 x + 4 y + 3 z = - 4$ with $- 8 - y + 2 z$ .

$- 2 (- 8 - y + 2 z) + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Simplify the left side.

Simplify $- 2 (- 8 - y + 2 z) + 4 y + 3 z$ .

Simplify each term.

Apply the distributive property.

$- 2 \cdot - 8 - 2 (- y) - 2 (2 z) + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Simplify.

Multiply $- 2$ by $- 8$ .

$16 - 2 (- y) - 2 (2 z) + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Multiply $- 1$ by $- 2$ .

$16 + 2 y - 2 (2 z) + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Multiply $2$ by $- 2$ .

$16 + 2 y - 4 z + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 2 y - 4 z + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 2 y - 4 z + 4 y + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Simplify by adding terms.

Add $2 y$ and $4 y$ .

$16 + 6 y - 4 z + 3 z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Add $- 4 z$ and $3 z$ .

$16 + 6 y - z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 6 y - z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 6 y - z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 6 y - z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$16 + 6 y - z = - 4$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Step 3

Solve for $y$ in $16 + 6 y - z = - 4$ .

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

Subtract $16$ from both sides of the equation.

$6 y - z = - 4 - 16$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Add $z$ to both sides of the equation.

$6 y = - 4 - 16 + z$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Subtract $16$ from $- 4$ .

$6 y = - 20 + z$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$6 y = - 20 + z$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Divide each term in $6 y = - 20 + z$ by $6$ and simplify.

Divide each term in $6 y = - 20 + z$ by $6$ .

$\frac{6 y}{6} = \frac{- 20}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 20}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Divide $y$ by $1$ .

$y = \frac{- 20}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{- 20}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{- 20}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 20$ and $6$ .

Factor $2$ out of $- 20$ .

$y = \frac{2 (- 10)}{6} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{2 \cdot - 10}{2 \cdot 3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Cancel the common factor.

$y = \frac{2 \cdot - 10}{2 \cdot 3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Rewrite the expression.

$y = \frac{- 10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{- 10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{- 10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$y = - \frac{10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = - \frac{10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = - \frac{10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = - \frac{10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

$y = - \frac{10}{3} + \frac{z}{6}$

$8 + 4 y + 2 z = 0$

$x = - 8 - y + 2 z$

Step 4

Replace all occurrences of $y$ with $- \frac{10}{3} + \frac{z}{6}$ in each equation.

Replace all occurrences of $y$ in $8 + 4 y + 2 z = 0$ with $- \frac{10}{3} + \frac{z}{6}$ .

$8 + 4 (- \frac{10}{3} + \frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Simplify the left side.

Simplify $8 + 4 (- \frac{10}{3} + \frac{z}{6}) + 2 z$ .

Simplify each term.

Apply the distributive property.

$8 + 4 (- \frac{10}{3}) + 4 (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Multiply $4 (- \frac{10}{3})$ .

Multiply $- 1$ by $4$ .

$8 - 4 (\frac{10}{3}) + 4 (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Combine $- 4$ and $\frac{10}{3}$ .

$8 + \frac{- 4 \cdot 10}{3} + 4 (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Multiply $- 4$ by $10$ .

$8 + \frac{- 40}{3} + 4 (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$8 + \frac{- 40}{3} + 4 (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$8 + \frac{- 40}{3} + 2 (2) (\frac{z}{6}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Factor $2$ out of $6$ .

$8 + \frac{- 40}{3} + 2 \cdot (2 (\frac{z}{2 \cdot 3})) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Cancel the common factor.

$8 + \frac{- 40}{3} + 2 \cdot (2 (\frac{z}{2 \cdot 3})) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Rewrite the expression.

$8 + \frac{- 40}{3} + 2 (\frac{z}{3}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$8 + \frac{- 40}{3} + 2 (\frac{z}{3}) + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

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

$8 + \frac{- 40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$8 - \frac{40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$8 - \frac{40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} - \frac{40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Combine $8$ and $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3} - \frac{40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 - 40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Simplify the numerator.

Multiply $8$ by $3$ .

$\frac{24 - 40}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Subtract $40$ from $24$ .

$\frac{- 16}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$\frac{- 16}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$- \frac{16}{3} + \frac{2 z}{3} + 2 z = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{16}{3} + \frac{2 z}{3} + 2 z \cdot \frac{3}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

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

$- \frac{16}{3} + \frac{2 z}{3} + \frac{2 z \cdot 3}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$- \frac{16}{3} + \frac{2 z + 2 z \cdot 3}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$\frac{- 16 + 2 z + 2 z \cdot 3}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Multiply $3$ by $2$ .

$\frac{- 16 + 2 z + 6 z}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Add $2 z$ and $6 z$ .

$\frac{- 16 + 8 z}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Factor $8$ out of $- 16 + 8 z$ .

Factor $8$ out of $- 16$ .

$\frac{8 \cdot - 2 + 8 z}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Factor $8$ out of $8 \cdot - 2 + 8 z$ .

$\frac{8 (- 2 + z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$\frac{8 (- 2 + z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{8 (- 1 \cdot 2 + z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Factor $- 1$ out of $z$ .

$\frac{8 (- 1 \cdot 2 - 1 (- z))}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Factor $- 1$ out of $- 1 (2) - 1 (- z)$ .

$\frac{8 (- 1 (2 - z))}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 - y + 2 z$

Replace all occurrences of $y$ in $x = - 8 - y + 2 z$ with $- \frac{10}{3} + \frac{z}{6}$ .

$x = - 8 - (- \frac{10}{3} + \frac{z}{6}) + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the right side.

Simplify $- 8 - (- \frac{10}{3} + \frac{z}{6}) + 2 z$ .

Simplify each term.

Apply the distributive property.

$x = - 8 + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Multiply $- - \frac{10}{3}$ .

Multiply $- 1$ by $- 1$ .

$x = - 8 + 1 (\frac{10}{3}) - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Multiply $\frac{10}{3}$ by $1$ .

$x = - 8 + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 8 + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - 8 \cdot \frac{3}{3} + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Combine $- 8$ and $\frac{3}{3}$ .

$x = \frac{- 8 \cdot 3}{3} + \frac{10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Combine the numerators over the common denominator.

$x = \frac{- 8 \cdot 3 + 10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the numerator.

Multiply $- 8$ by $3$ .

$x = \frac{- 24 + 10}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Add $- 24$ and $10$ .

$x = \frac{- 14}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = \frac{- 14}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Move the negative in front of the fraction.

$x = - \frac{14}{3} - \frac{z}{6} + 2 z$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = - \frac{14}{3} - \frac{z}{6} + 2 z \cdot \frac{6}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify terms.

Combine $2 z$ and $\frac{6}{6}$ .

$x = - \frac{14}{3} - \frac{z}{6} + \frac{2 z \cdot 6}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Combine the numerators over the common denominator.

$x = - \frac{14}{3} + \frac{- z + 2 z \cdot 6}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{- z + 2 z \cdot 6}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify each term.

Simplify the numerator.

Factor $z$ out of $- z + 2 z \cdot 6$ .

Factor $z$ out of $- z$ .

$x = - \frac{14}{3} + \frac{z \cdot - 1 + 2 z \cdot 6}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Factor $z$ out of $2 z \cdot 6$ .

$x = - \frac{14}{3} + \frac{z \cdot - 1 + z (2 \cdot 6)}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Factor $z$ out of $z \cdot - 1 + z (2 \cdot 6)$ .

$x = - \frac{14}{3} + \frac{z (- 1 + 2 \cdot 6)}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{z (- 1 + 2 \cdot 6)}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Multiply $2$ by $6$ .

$x = - \frac{14}{3} + \frac{z (- 1 + 12)}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Add $- 1$ and $12$ .

$x = - \frac{14}{3} + \frac{z \cdot 11}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{z \cdot 11}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Move $11$ to the left of $z$ .

$x = - \frac{14}{3} + \frac{11 z}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$- \frac{8 (2 - z)}{3} = 0$

$y = - \frac{10}{3} + \frac{z}{6}$

Step 5

Solve for $z$ in $- \frac{8 (2 - z)}{3} = 0$ .

Set the numerator equal to zero.

$8 (2 - z) = 0$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Solve the equation for $z$ .

Divide each term in $8 (2 - z) = 0$ by $8$ and simplify.

Divide each term in $8 (2 - z) = 0$ by $8$ .

$\frac{8 (2 - z)}{8} = \frac{0}{8}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (2 - z)}{8} = \frac{0}{8}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Divide $2 - z$ by $1$ .

$2 - z = \frac{0}{8}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$2 - z = \frac{0}{8}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$2 - z = \frac{0}{8}$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the right side.

Divide $0$ by $8$ .

$2 - z = 0$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$2 - z = 0$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$2 - z = 0$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Subtract $2$ from both sides of the equation.

$- z = - 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

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

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

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

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Divide $z$ by $1$ .

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

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

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

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$z = 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$z = 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$z = 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$z = 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

$z = 2$

$x = - \frac{14}{3} + \frac{11 z}{6}$

$y = - \frac{10}{3} + \frac{z}{6}$

Step 6

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

Replace all occurrences of $z$ in $x = - \frac{14}{3} + \frac{11 z}{6}$ with $2$ .

$x = - \frac{14}{3} + \frac{11 (2)}{6}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the right side.

Simplify $- \frac{14}{3} + \frac{11 (2)}{6}$ .

Multiply $11$ by $2$ .

$x = - \frac{14}{3} + \frac{22}{6}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Cancel the common factor of $22$ and $6$ .

Factor $2$ out of $22$ .

$x = - \frac{14}{3} + \frac{2 (11)}{6}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = - \frac{14}{3} + \frac{2 \cdot 11}{2 \cdot 3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Cancel the common factor.

$x = - \frac{14}{3} + \frac{2 \cdot 11}{2 \cdot 3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Rewrite the expression.

$x = - \frac{14}{3} + \frac{11}{3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11}{3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - \frac{14}{3} + \frac{11}{3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Combine the numerators over the common denominator.

$x = \frac{- 14 + 11}{3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Simplify the expression.

Add $- 14$ and $11$ .

$x = \frac{- 3}{3}$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Divide $- 3$ by $3$ .

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{z}{6}$

Replace all occurrences of $z$ in $y = - \frac{10}{3} + \frac{z}{6}$ with $2$ .

$y = - \frac{10}{3} + \frac{2}{6}$

$x = - 1$

$z = 2$

Simplify the right side.

Simplify $- \frac{10}{3} + \frac{2}{6}$ .

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$y = - \frac{10}{3} + \frac{2 (1)}{6}$

$x = - 1$

$z = 2$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = - \frac{10}{3} + \frac{2 \cdot 1}{2 \cdot 3}$

$x = - 1$

$z = 2$

Cancel the common factor.

$y = - \frac{10}{3} + \frac{2 \cdot 1}{2 \cdot 3}$

$x = - 1$

$z = 2$

Rewrite the expression.

$y = - \frac{10}{3} + \frac{1}{3}$

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{1}{3}$

$x = - 1$

$z = 2$

$y = - \frac{10}{3} + \frac{1}{3}$

$x = - 1$

$z = 2$

Combine the numerators over the common denominator.

$y = \frac{- 10 + 1}{3}$

$x = - 1$

$z = 2$

Simplify the expression.

Add $- 10$ and $1$ .

$y = \frac{- 9}{3}$

$x = - 1$

$z = 2$

Divide $- 9$ by $3$ .

$y = - 3$

$x = - 1$

$z = 2$

$y = - 3$

$x = - 1$

$z = 2$

$y = - 3$

$x = - 1$

$z = 2$

$y = - 3$

$x = - 1$

$z = 2$

$y = - 3$

$x = - 1$

$z = 2$

Step 7

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

$(- 1, - 3, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 1, - 3, 2)$

Equation Form:

$x = - 1, y = - 3, z = 2$

Solve the System of Equations -x+3y+4z=0 x+y-2z=-8 -2x+4y+3z=-4

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