Solve the System of Equations 4x-y-8z=5 x-y-8z=-4 x+y+7z=9

$4 x - y - 8 z = 5$ $x - y - 8 z = - 4$ $x + y + 7 z = 9$

Step 1

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

Add $y$ to both sides of the equation.

$x - 8 z = - 4 + y$

$4 x - y - 8 z = 5$

$x + y + 7 z = 9$

Add $8 z$ to both sides of the equation.

$x = - 4 + y + 8 z$

$4 x - y - 8 z = 5$

$x + y + 7 z = 9$

$x = - 4 + y + 8 z$

$4 x - y - 8 z = 5$

$x + y + 7 z = 9$

Step 2

Replace all occurrences of $x$ with $- 4 + y + 8 z$ in each equation.

Replace all occurrences of $x$ in $4 x - y - 8 z = 5$ with $- 4 + y + 8 z$ .

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

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$4 \cdot - 4 + 4 y + 4 (8 z) - y - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Simplify.

Multiply $4$ by $- 4$ .

$- 16 + 4 y + 4 (8 z) - y - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Multiply $8$ by $4$ .

$- 16 + 4 y + 32 z - y - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

$- 16 + 4 y + 32 z - y - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

$- 16 + 4 y + 32 z - y - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Simplify by adding terms.

Subtract $y$ from $4 y$ .

$- 16 + 3 y + 32 z - 8 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Subtract $8 z$ from $32 z$ .

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$x + y + 7 z = 9$

Replace all occurrences of $x$ in $x + y + 7 z = 9$ with $- 4 + y + 8 z$ .

$(- 4 + y + 8 z) + y + 7 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Simplify the left side.

Simplify $(- 4 + y + 8 z) + y + 7 z$ .

Add $y$ and $y$ .

$- 4 + 2 y + 8 z + 7 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Add $8 z$ and $7 z$ .

$- 4 + 2 y + 15 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$- 4 + 2 y + 15 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$- 4 + 2 y + 15 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$- 4 + 2 y + 15 z = 9$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Step 3

Solve for $y$ in $- 4 + 2 y + 15 z = 9$ .

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

Add $4$ to both sides of the equation.

$2 y + 15 z = 9 + 4$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Subtract $15 z$ from both sides of the equation.

$2 y = 9 + 4 - 15 z$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Add $9$ and $4$ .

$2 y = 13 - 15 z$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$2 y = 13 - 15 z$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Divide each term in $2 y = 13 - 15 z$ by $2$ and simplify.

Divide each term in $2 y = 13 - 15 z$ by $2$ .

$\frac{2 y}{2} = \frac{13}{2} + \frac{- 15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{13}{2} + \frac{- 15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Divide $y$ by $1$ .

$y = \frac{13}{2} + \frac{- 15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$y = \frac{13}{2} + \frac{- 15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$y = \frac{13}{2} + \frac{- 15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{13}{2} - \frac{15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$y = \frac{13}{2} - \frac{15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$y = \frac{13}{2} - \frac{15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

$y = \frac{13}{2} - \frac{15 z}{2}$

$- 16 + 3 y + 24 z = 5$

$x = - 4 + y + 8 z$

Step 4

Replace all occurrences of $y$ with $\frac{13}{2} - \frac{15 z}{2}$ in each equation.

Replace all occurrences of $y$ in $- 16 + 3 y + 24 z = 5$ with $\frac{13}{2} - \frac{15 z}{2}$ .

$- 16 + 3 (\frac{13}{2} - \frac{15 z}{2}) + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Simplify the left side.

Simplify $- 16 + 3 (\frac{13}{2} - \frac{15 z}{2}) + 24 z$ .

Simplify each term.

Apply the distributive property.

$- 16 + 3 (\frac{13}{2}) + 3 (- \frac{15 z}{2}) + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Multiply $3 (\frac{13}{2})$ .

Combine $3$ and $\frac{13}{2}$ .

$- 16 + \frac{3 \cdot 13}{2} + 3 (- \frac{15 z}{2}) + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Multiply $3$ by $13$ .

$- 16 + \frac{39}{2} + 3 (- \frac{15 z}{2}) + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$- 16 + \frac{39}{2} + 3 (- \frac{15 z}{2}) + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Multiply $3 (- \frac{15 z}{2})$ .

Multiply $- 1$ by $3$ .

$- 16 + \frac{39}{2} - 3 \frac{15 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine $- 3$ and $\frac{15 z}{2}$ .

$- 16 + \frac{39}{2} + \frac{- 3 (15 z)}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Multiply $15$ by $- 3$ .

$- 16 + \frac{39}{2} + \frac{- 45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$- 16 + \frac{39}{2} + \frac{- 45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Move the negative in front of the fraction.

$- 16 + \frac{39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$- 16 + \frac{39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 16 \cdot \frac{2}{2} + \frac{39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine $- 16$ and $\frac{2}{2}$ .

$\frac{- 16 \cdot 2}{2} + \frac{39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine the numerators over the common denominator.

$\frac{- 16 \cdot 2 + 39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Simplify the numerator.

Multiply $- 16$ by $2$ .

$\frac{- 32 + 39}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Add $- 32$ and $39$ .

$\frac{7}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$\frac{7}{2} - \frac{45 z}{2} + 24 z = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

To write $24 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7}{2} - \frac{45 z}{2} + 24 z \cdot \frac{2}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine $24 z$ and $\frac{2}{2}$ .

$\frac{7}{2} - \frac{45 z}{2} + \frac{24 z \cdot 2}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine the numerators over the common denominator.

$\frac{7}{2} + \frac{- 45 z + 24 z \cdot 2}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Combine the numerators over the common denominator.

$\frac{7 - 45 z + 24 z \cdot 2}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Multiply $2$ by $24$ .

$\frac{7 - 45 z + 48 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Add $- 45 z$ and $48 z$ .

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + y + 8 z$

Replace all occurrences of $y$ in $x = - 4 + y + 8 z$ with $\frac{13}{2} - \frac{15 z}{2}$ .

$x = - 4 + \frac{13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify $x = - 4 + \frac{13}{2} - \frac{15 z}{2} + 8 z$ .

Simplify the left side.

Remove parentheses.

$x = - 4 + \frac{13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = - 4 + \frac{13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the right side.

Simplify $- 4 + \frac{13}{2} - \frac{15 z}{2} + 8 z$ .

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - 4 \cdot \frac{2}{2} + \frac{13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Combine $- 4$ and $\frac{2}{2}$ .

$x = \frac{- 4 \cdot 2}{2} + \frac{13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 4 \cdot 2 + 13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the numerator.

Multiply $- 4$ by $2$ .

$x = \frac{- 8 + 13}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Add $- 8$ and $13$ .

$x = \frac{5}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5}{2} - \frac{15 z}{2} + 8 z$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

To write $8 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{5}{2} - \frac{15 z}{2} + 8 z \cdot \frac{2}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Combine $8 z$ and $\frac{2}{2}$ .

$x = \frac{5}{2} - \frac{15 z}{2} + \frac{8 z \cdot 2}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Combine the numerators over the common denominator.

$x = \frac{5}{2} + \frac{- 15 z + 8 z \cdot 2}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Combine the numerators over the common denominator.

$x = \frac{5 - 15 z + 8 z \cdot 2}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Multiply $2$ by $8$ .

$x = \frac{5 - 15 z + 16 z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Add $- 15 z$ and $16 z$ .

$x = \frac{5 + z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5 + z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5 + z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5 + z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5 + z}{2}$

$\frac{7 + 3 z}{2} = 5$

$y = \frac{13}{2} - \frac{15 z}{2}$

Step 5

Solve for $z$ in $\frac{7 + 3 z}{2} = 5$ .

Multiply both sides by $2$ .

$\frac{7 + 3 z}{2} \cdot 2 = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify.

Simplify the left side.

Simplify $\frac{7 + 3 z}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{7 + 3 z}{2} \cdot 2 = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Rewrite the expression.

$7 + 3 z = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$7 + 3 z = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Reorder $7$ and $3 z$ .

$3 z + 7 = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$3 z + 7 = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$3 z + 7 = 5 \cdot 2$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the right side.

Multiply $5$ by $2$ .

$3 z + 7 = 10$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$3 z + 7 = 10$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$3 z + 7 = 10$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Solve for $z$ .

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

Subtract $7$ from both sides of the equation.

$3 z = 10 - 7$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Subtract $7$ from $10$ .

$3 z = 3$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$3 z = 3$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Divide each term in $3 z = 3$ by $3$ and simplify.

Divide each term in $3 z = 3$ by $3$ .

$\frac{3 z}{3} = \frac{3}{3}$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z}{3} = \frac{3}{3}$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Divide $z$ by $1$ .

$z = \frac{3}{3}$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = \frac{3}{3}$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = \frac{3}{3}$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the right side.

Divide $3$ by $3$ .

$z = 1$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = 1$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = 1$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = 1$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

$z = 1$

$x = \frac{5 + z}{2}$

$y = \frac{13}{2} - \frac{15 z}{2}$

Step 6

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

Replace all occurrences of $z$ in $x = \frac{5 + z}{2}$ with $1$ .

$x = \frac{5 + 1}{2}$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify $x = \frac{5 + 1}{2}$ .

Simplify the left side.

Remove parentheses.

$x = \frac{5 + 1}{2}$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = \frac{5 + 1}{2}$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

Simplify the right side.

Simplify $\frac{5 + 1}{2}$ .

Add $5$ and $1$ .

$x = \frac{6}{2}$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

Divide $6$ by $2$ .

$x = 3$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = 3$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = 3$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

$x = 3$

$z = 1$

$y = \frac{13}{2} - \frac{15 z}{2}$

Replace all occurrences of $z$ in $y = \frac{13}{2} - \frac{15 z}{2}$ with $1$ .

$y = \frac{13}{2} - \frac{15 (1)}{2}$

$x = 3$

$z = 1$

Simplify the right side.

Simplify $\frac{13}{2} - \frac{15 (1)}{2}$ .

Combine the numerators over the common denominator.

$y = \frac{13 - 15 \cdot 1}{2}$

$x = 3$

$z = 1$

Simplify the expression.

Multiply $- 15$ by $1$ .

$y = \frac{13 - 15}{2}$

$x = 3$

$z = 1$

Subtract $15$ from $13$ .

$y = \frac{- 2}{2}$

$x = 3$

$z = 1$

Divide $- 2$ by $2$ .

$y = - 1$

$x = 3$

$z = 1$

$y = - 1$

$x = 3$

$z = 1$

$y = - 1$

$x = 3$

$z = 1$

$y = - 1$

$x = 3$

$z = 1$

$y = - 1$

$x = 3$

$z = 1$

Step 7

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

$(3, - 1, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, - 1, 1)$

Equation Form:

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

Solve the System of Equations 4x-y-8z=5 x-y-8z=-4 x+y+7z=9

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