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

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

Step 1

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

Subtract $y$ from both sides of the equation.

$x - 2 z = - 8 - y$

$3 x - y + 4 z = 0$

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

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

$x = - 8 - y + 2 z$

$3 x - y + 4 z = 0$

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

$x = - 8 - y + 2 z$

$3 x - y + 4 z = 0$

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

Step 2

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

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

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

$x = - 8 - y + 2 z$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$x = - 8 - y + 2 z$

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

Simplify.

Multiply $3$ by $- 8$ .

$- 24 + 3 (- y) + 3 (2 z) - y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Multiply $- 1$ by $3$ .

$- 24 - 3 y + 3 (2 z) - y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Multiply $2$ by $3$ .

$- 24 - 3 y + 6 z - y + 4 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 3 y + 6 z - y + 4 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 3 y + 6 z - y + 4 z = 0$

$x = - 8 - y + 2 z$

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

Simplify by adding terms.

Subtract $y$ from $- 3 y$ .

$- 24 - 4 y + 6 z + 4 z = 0$

$x = - 8 - y + 2 z$

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

Add $6 z$ and $4 z$ .

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

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

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Simplify.

Multiply $4$ by $- 8$ .

$- 32 + 4 (- y) + 4 (2 z) - 2 y + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Multiply $- 1$ by $4$ .

$- 32 - 4 y + 4 (2 z) - 2 y + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Multiply $2$ by $4$ .

$- 32 - 4 y + 8 z - 2 y + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 4 y + 8 z - 2 y + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 4 y + 8 z - 2 y + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Simplify by adding terms.

Subtract $2 y$ from $- 4 y$ .

$- 32 - 6 y + 8 z + 3 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Add $8 z$ and $3 z$ .

$- 32 - 6 y + 11 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 6 y + 11 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 6 y + 11 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 6 y + 11 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 32 - 6 y + 11 z = - 4$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Step 3

Solve for $y$ in $- 32 - 6 y + 11 z = - 4$ .

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

Add $32$ to both sides of the equation.

$- 6 y + 11 z = - 4 + 32$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Subtract $11 z$ from both sides of the equation.

$- 6 y = - 4 + 32 - 11 z$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Add $- 4$ and $32$ .

$- 6 y = 28 - 11 z$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$- 6 y = 28 - 11 z$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Divide each term in $- 6 y = 28 - 11 z$ by $- 6$ and simplify.

Divide each term in $- 6 y = 28 - 11 z$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{28}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{28}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Divide $y$ by $1$ .

$y = \frac{28}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{28}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

$y = \frac{28}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Simplify the right side.

Simplify each term.

Cancel the common factor of $28$ and $- 6$ .

Factor $2$ out of $28$ .

$y = \frac{2 (14)}{- 6} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$y = \frac{2 \cdot 14}{2 \cdot - 3} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Cancel the common factor.

$y = \frac{2 \cdot 14}{2 \cdot - 3} + \frac{- 11 z}{- 6}$

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Rewrite the expression.

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Dividing two negative values results in a positive value.

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

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

$- 24 - 4 y + 10 z = 0$

$x = - 8 - y + 2 z$

Step 4

Replace all occurrences of $y$ with $- \frac{14}{3} + \frac{11 z}{6}$ in each equation.

Replace all occurrences of $y$ in $- 24 - 4 y + 10 z = 0$ with $- \frac{14}{3} + \frac{11 z}{6}$ .

$- 24 - 4 (- \frac{14}{3} + \frac{11 z}{6}) + 10 z = 0$

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

$x = - 8 - y + 2 z$

Simplify the left side.

Simplify $- 24 - 4 (- \frac{14}{3} + \frac{11 z}{6}) + 10 z$ .

Simplify each term.

Apply the distributive property.

$- 24 - 4 (- \frac{14}{3}) - 4 \frac{11 z}{6} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Multiply $- 4 (- \frac{14}{3})$ .

Multiply $- 1$ by $- 4$ .

$- 24 + 4 (\frac{14}{3}) - 4 \frac{11 z}{6} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Combine $4$ and $\frac{14}{3}$ .

$- 24 + \frac{4 \cdot 14}{3} - 4 \frac{11 z}{6} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Multiply $4$ by $14$ .

$- 24 + \frac{56}{3} - 4 \frac{11 z}{6} + 10 z = 0$

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

$x = - 8 - y + 2 z$

$- 24 + \frac{56}{3} - 4 \frac{11 z}{6} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

$- 24 + \frac{56}{3} + 2 (- 2) (\frac{11 z}{6}) + 10 z = 0$

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

$x = - 8 - y + 2 z$

Factor $2$ out of $6$ .

$- 24 + \frac{56}{3} + 2 \cdot (- 2 \frac{11 z}{2 \cdot 3}) + 10 z = 0$

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

$x = - 8 - y + 2 z$

Cancel the common factor.

$- 24 + \frac{56}{3} + 2 \cdot (- 2 \frac{11 z}{2 \cdot 3}) + 10 z = 0$

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

$x = - 8 - y + 2 z$

Rewrite the expression.

$- 24 + \frac{56}{3} - 2 \frac{11 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

$- 24 + \frac{56}{3} - 2 \frac{11 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Combine $- 2$ and $\frac{11 z}{3}$ .

$- 24 + \frac{56}{3} + \frac{- 2 (11 z)}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Multiply $11$ by $- 2$ .

$- 24 + \frac{56}{3} + \frac{- 22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$- 24 + \frac{56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

$- 24 + \frac{56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

To write $- 24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 24 \cdot \frac{3}{3} + \frac{56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Combine $- 24$ and $\frac{3}{3}$ .

$\frac{- 24 \cdot 3}{3} + \frac{56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$\frac{- 24 \cdot 3 + 56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Simplify the numerator.

Multiply $- 24$ by $3$ .

$\frac{- 72 + 56}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Add $- 72$ and $56$ .

$\frac{- 16}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

$\frac{- 16}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

$- \frac{16}{3} - \frac{22 z}{3} + 10 z = 0$

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

$x = - 8 - y + 2 z$

To write $10 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{16}{3} - \frac{22 z}{3} + 10 z \cdot \frac{3}{3} = 0$

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

$x = - 8 - y + 2 z$

Combine $10 z$ and $\frac{3}{3}$ .

$- \frac{16}{3} - \frac{22 z}{3} + \frac{10 z \cdot 3}{3} = 0$

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

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$- \frac{16}{3} + \frac{- 22 z + 10 z \cdot 3}{3} = 0$

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

$x = - 8 - y + 2 z$

Combine the numerators over the common denominator.

$\frac{- 16 - 22 z + 10 z \cdot 3}{3} = 0$

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

$x = - 8 - y + 2 z$

Multiply $3$ by $10$ .

$\frac{- 16 - 22 z + 30 z}{3} = 0$

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

$x = - 8 - y + 2 z$

Add $- 22 z$ and $30 z$ .

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

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

$x = - 8 - y + 2 z$

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

Factor $8$ out of $- 16$ .

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

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

$x = - 8 - y + 2 z$

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

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

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

$x = - 8 - y + 2 z$

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

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

$x = - 8 - y + 2 z$

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

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

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

$x = - 8 - y + 2 z$

Factor $- 1$ out of $z$ .

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

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

$x = - 8 - y + 2 z$

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

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

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

$x = - 8 - y + 2 z$

Move the negative in front of the fraction.

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

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

$x = - 8 - y + 2 z$

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

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

$x = - 8 - y + 2 z$

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

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

$x = - 8 - y + 2 z$

Replace all occurrences of $y$ in $x = - 8 - y + 2 z$ with $- \frac{14}{3} + \frac{11 z}{6}$ .

$x = - 8 - (- \frac{14}{3} + \frac{11 z}{6}) + 2 z$

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

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

Simplify the right side.

Simplify $- 8 - (- \frac{14}{3} + \frac{11 z}{6}) + 2 z$ .

Simplify each term.

Apply the distributive property.

$x = - 8 + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

Multiply $- - \frac{14}{3}$ .

Multiply $- 1$ by $- 1$ .

$x = - 8 + 1 (\frac{14}{3}) - \frac{11 z}{6} + 2 z$

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

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

Multiply $\frac{14}{3}$ by $1$ .

$x = - 8 + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

$x = - 8 + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

$x = - 8 + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

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

$x = - 8 \cdot \frac{3}{3} + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

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

$x = \frac{- 8 \cdot 3}{3} + \frac{14}{3} - \frac{11 z}{6} + 2 z$

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

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

Combine the numerators over the common denominator.

$x = \frac{- 8 \cdot 3 + 14}{3} - \frac{11 z}{6} + 2 z$

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

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

Simplify the numerator.

Multiply $- 8$ by $3$ .

$x = \frac{- 24 + 14}{3} - \frac{11 z}{6} + 2 z$

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

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

Add $- 24$ and $14$ .

$x = \frac{- 10}{3} - \frac{11 z}{6} + 2 z$

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

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

$x = \frac{- 10}{3} - \frac{11 z}{6} + 2 z$

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

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

Move the negative in front of the fraction.

$x = - \frac{10}{3} - \frac{11 z}{6} + 2 z$

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

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

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

$x = - \frac{10}{3} - \frac{11 z}{6} + 2 z \cdot \frac{6}{6}$

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

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

Simplify terms.

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

$x = - \frac{10}{3} - \frac{11 z}{6} + \frac{2 z \cdot 6}{6}$

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

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

Combine the numerators over the common denominator.

$x = - \frac{10}{3} + \frac{- 11 z + 2 z \cdot 6}{6}$

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

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

$x = - \frac{10}{3} + \frac{- 11 z + 2 z \cdot 6}{6}$

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

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

Simplify each term.

Simplify the numerator.

Factor $z$ out of $- 11 z + 2 z \cdot 6$ .

Factor $z$ out of $- 11 z$ .

$x = - \frac{10}{3} + \frac{z \cdot - 11 + 2 z \cdot 6}{6}$

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

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

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

$x = - \frac{10}{3} + \frac{z \cdot - 11 + z (2 \cdot 6)}{6}$

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

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

Factor $z$ out of $z \cdot - 11 + z (2 \cdot 6)$ .

$x = - \frac{10}{3} + \frac{z (- 11 + 2 \cdot 6)}{6}$

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

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

$x = - \frac{10}{3} + \frac{z (- 11 + 2 \cdot 6)}{6}$

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

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

Multiply $2$ by $6$ .

$x = - \frac{10}{3} + \frac{z (- 11 + 12)}{6}$

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

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

Add $- 11$ and $12$ .

$x = - \frac{10}{3} + \frac{z \cdot 1}{6}$

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

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

$x = - \frac{10}{3} + \frac{z \cdot 1}{6}$

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

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

Multiply $z$ by $1$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 5

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

Set the numerator equal to zero.

$8 (2 - z) = 0$

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

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

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 $8$ .

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

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

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

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

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

Divide $2 - z$ by $1$ .

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

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

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

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

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

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

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

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

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

Simplify the right side.

Divide $0$ by $8$ .

$2 - z = 0$

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

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

$2 - z = 0$

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

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

$2 - z = 0$

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

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

Subtract $2$ from both sides of the equation.

$- z = - 2$

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

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

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

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

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Divide $z$ by $1$ .

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

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

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

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

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

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

Simplify the right side.

Divide $- 2$ by $- 1$ .

$z = 2$

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

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

$z = 2$

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

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

$z = 2$

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

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

$z = 2$

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

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

$z = 2$

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

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

Step 6

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

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

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

$z = 2$

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

Simplify the right side.

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

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

Factor $2$ out of $2$ .

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

$z = 2$

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

$z = 2$

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

Cancel the common factor.

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

$z = 2$

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

Rewrite the expression.

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

$z = 2$

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

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

$z = 2$

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

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

$z = 2$

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

Combine the numerators over the common denominator.

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

$z = 2$

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

Simplify the expression.

Add $- 10$ and $1$ .

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

$z = 2$

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

Divide $- 9$ by $3$ .

$x = - 3$

$z = 2$

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

$x = - 3$

$z = 2$

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

$x = - 3$

$z = 2$

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

$x = - 3$

$z = 2$

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

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

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

$x = - 3$

$z = 2$

Simplify the right side.

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

Multiply $11$ by $2$ .

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

$x = - 3$

$z = 2$

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

Factor $2$ out of $22$ .

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

$x = - 3$

$z = 2$

Cancel the common factors.

Factor $2$ out of $6$ .

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

$x = - 3$

$z = 2$

Cancel the common factor.

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

$x = - 3$

$z = 2$

Rewrite the expression.

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

$x = - 3$

$z = 2$

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

$x = - 3$

$z = 2$

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

$x = - 3$

$z = 2$

Combine the numerators over the common denominator.

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

$x = - 3$

$z = 2$

Simplify the expression.

Add $- 14$ and $11$ .

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

$x = - 3$

$z = 2$

Divide $- 3$ by $3$ .

$y = - 1$

$x = - 3$

$z = 2$

$y = - 1$

$x = - 3$

$z = 2$

$y = - 1$

$x = - 3$

$z = 2$

$y = - 1$

$x = - 3$

$z = 2$

$y = - 1$

$x = - 3$

$z = 2$

Step 7

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

$(- 3, - 1, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 3, - 1, 2)$

Equation Form:

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

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

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