Solve the System of Equations -5x+10y-4z=10 x+4y-4z=-8 x-7y+4z=3

$- 5 x + 10 y - 4 z = 10$ $x + 4 y - 4 z = - 8$ $x - 7 y + 4 z = 3$

Step 1

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

Subtract $4 y$ from both sides of the equation.

$x - 4 z = - 8 - 4 y$

$- 5 x + 10 y - 4 z = 10$

$x - 7 y + 4 z = 3$

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

$x = - 8 - 4 y + 4 z$

$- 5 x + 10 y - 4 z = 10$

$x - 7 y + 4 z = 3$

$x = - 8 - 4 y + 4 z$

$- 5 x + 10 y - 4 z = 10$

$x - 7 y + 4 z = 3$

Step 2

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

Replace all occurrences of $x$ in $- 5 x + 10 y - 4 z = 10$ with $- 8 - 4 y + 4 z$ .

$- 5 (- 8 - 4 y + 4 z) + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Simplify the left side.

Simplify $- 5 (- 8 - 4 y + 4 z) + 10 y - 4 z$ .

Simplify each term.

Apply the distributive property.

$- 5 \cdot - 8 - 5 (- 4 y) - 5 (4 z) + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Simplify.

Multiply $- 5$ by $- 8$ .

$40 - 5 (- 4 y) - 5 (4 z) + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Multiply $- 4$ by $- 5$ .

$40 + 20 y - 5 (4 z) + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Multiply $4$ by $- 5$ .

$40 + 20 y - 20 z + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

$40 + 20 y - 20 z + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

$40 + 20 y - 20 z + 10 y - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Simplify by adding terms.

Add $20 y$ and $10 y$ .

$40 + 30 y - 20 z - 4 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Subtract $4 z$ from $- 20 z$ .

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$x - 7 y + 4 z = 3$

Replace all occurrences of $x$ in $x - 7 y + 4 z = 3$ with $- 8 - 4 y + 4 z$ .

$(- 8 - 4 y + 4 z) - 7 y + 4 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Simplify the left side.

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

Subtract $7 y$ from $- 4 y$ .

$- 8 - 11 y + 4 z + 4 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Add $4 z$ and $4 z$ .

$- 8 - 11 y + 8 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$- 8 - 11 y + 8 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$- 8 - 11 y + 8 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$- 8 - 11 y + 8 z = 3$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Step 3

Solve for $y$ in $- 8 - 11 y + 8 z = 3$ .

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

Add $8$ to both sides of the equation.

$- 11 y + 8 z = 3 + 8$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Subtract $8 z$ from both sides of the equation.

$- 11 y = 3 + 8 - 8 z$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Add $3$ and $8$ .

$- 11 y = 11 - 8 z$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$- 11 y = 11 - 8 z$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Divide each term in $- 11 y = 11 - 8 z$ by $- 11$ and simplify.

Divide each term in $- 11 y = 11 - 8 z$ by $- 11$ .

$\frac{- 11 y}{- 11} = \frac{11}{- 11} + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Simplify the left side.

Cancel the common factor of $- 11$ .

Cancel the common factor.

$\frac{- 11 y}{- 11} = \frac{11}{- 11} + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Divide $y$ by $1$ .

$y = \frac{11}{- 11} + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = \frac{11}{- 11} + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = \frac{11}{- 11} + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Simplify the right side.

Simplify each term.

Divide $11$ by $- 11$ .

$y = - 1 + \frac{- 8 z}{- 11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Dividing two negative values results in a positive value.

$y = - 1 + \frac{8 z}{11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = - 1 + \frac{8 z}{11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = - 1 + \frac{8 z}{11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = - 1 + \frac{8 z}{11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

$y = - 1 + \frac{8 z}{11}$

$40 + 30 y - 24 z = 10$

$x = - 8 - 4 y + 4 z$

Step 4

Replace all occurrences of $y$ with $- 1 + \frac{8 z}{11}$ in each equation.

Replace all occurrences of $y$ in $40 + 30 y - 24 z = 10$ with $- 1 + \frac{8 z}{11}$ .

$40 + 30 (- 1 + \frac{8 z}{11}) - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Simplify the left side.

Simplify $40 + 30 (- 1 + \frac{8 z}{11}) - 24 z$ .

Simplify each term.

Apply the distributive property.

$40 + 30 \cdot - 1 + 30 (\frac{8 z}{11}) - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Multiply $30$ by $- 1$ .

$40 - 30 + 30 (\frac{8 z}{11}) - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Multiply $30 \frac{8 z}{11}$ .

Combine $30$ and $\frac{8 z}{11}$ .

$40 - 30 + \frac{30 (8 z)}{11} - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Multiply $8$ by $30$ .

$40 - 30 + \frac{240 z}{11} - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$40 - 30 + \frac{240 z}{11} - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$40 - 30 + \frac{240 z}{11} - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Subtract $30$ from $40$ .

$10 + \frac{240 z}{11} - 24 z = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

To write $- 24 z$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$10 + \frac{240 z}{11} - 24 z \cdot \frac{11}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Simplify terms.

Combine $- 24 z$ and $\frac{11}{11}$ .

$10 + \frac{240 z}{11} + \frac{- 24 z \cdot 11}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Combine the numerators over the common denominator.

$10 + \frac{240 z - 24 z \cdot 11}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 + \frac{240 z - 24 z \cdot 11}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Simplify each term.

Simplify the numerator.

Factor $24 z$ out of $240 z - 24 z \cdot 11$ .

Factor $24 z$ out of $240 z$ .

$10 + \frac{24 z (10) - 24 z \cdot 11}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Factor $24 z$ out of $- 24 z \cdot 11$ .

$10 + \frac{24 z (10) + 24 z (- 1 \cdot 11)}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Factor $24 z$ out of $24 z (10) + 24 z (- 1 \cdot 11)$ .

$10 + \frac{24 z (10 - 1 \cdot 11)}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 + \frac{24 z (10 - 1 \cdot 11)}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Multiply $- 1$ by $11$ .

$10 + \frac{24 z (10 - 11)}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Subtract $11$ from $10$ .

$10 + \frac{24 z \cdot - 1}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Multiply $- 1$ by $24$ .

$10 + \frac{- 24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 + \frac{- 24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Move the negative in front of the fraction.

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 - 4 y + 4 z$

Replace all occurrences of $y$ in $x = - 8 - 4 y + 4 z$ with $- 1 + \frac{8 z}{11}$ .

$x = - 8 - 4 (- 1 + \frac{8 z}{11}) + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Simplify the right side.

Simplify $- 8 - 4 (- 1 + \frac{8 z}{11}) + 4 z$ .

Simplify each term.

Apply the distributive property.

$x = - 8 - 4 \cdot - 1 - 4 \frac{8 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Multiply $- 4$ by $- 1$ .

$x = - 8 + 4 - 4 \frac{8 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Multiply $- 4 \frac{8 z}{11}$ .

Combine $- 4$ and $\frac{8 z}{11}$ .

$x = - 8 + 4 + \frac{- 4 (8 z)}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Multiply $8$ by $- 4$ .

$x = - 8 + 4 + \frac{- 32 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 + 4 + \frac{- 32 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Move the negative in front of the fraction.

$x = - 8 + 4 - \frac{32 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 8 + 4 - \frac{32 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Add $- 8$ and $4$ .

$x = - 4 - \frac{32 z}{11} + 4 z$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

To write $4 z$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x = - 4 - \frac{32 z}{11} + 4 z \cdot \frac{11}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Simplify terms.

Combine $4 z$ and $\frac{11}{11}$ .

$x = - 4 - \frac{32 z}{11} + \frac{4 z \cdot 11}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Combine the numerators over the common denominator.

$x = - 4 + \frac{- 32 z + 4 z \cdot 11}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{- 32 z + 4 z \cdot 11}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Simplify the numerator.

Factor $4 z$ out of $- 32 z + 4 z \cdot 11$ .

Factor $4 z$ out of $- 32 z$ .

$x = - 4 + \frac{4 z (- 8) + 4 z \cdot 11}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Factor $4 z$ out of $4 z \cdot 11$ .

$x = - 4 + \frac{4 z (- 8) + 4 z (11)}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Factor $4 z$ out of $4 z (- 8) + 4 z (11)$ .

$x = - 4 + \frac{4 z (- 8 + 11)}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{4 z (- 8 + 11)}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Add $- 8$ and $11$ .

$x = - 4 + \frac{4 z \cdot 3}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Multiply $3$ by $4$ .

$x = - 4 + \frac{12 z}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{12 z}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{12 z}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{12 z}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

$x = - 4 + \frac{12 z}{11}$

$10 - \frac{24 z}{11} = 10$

$y = - 1 + \frac{8 z}{11}$

Step 5

Solve for $z$ in $10 - \frac{24 z}{11} = 10$ .

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

Subtract $10$ from both sides of the equation.

$- \frac{24 z}{11} = 10 - 10$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Subtract $10$ from $10$ .

$- \frac{24 z}{11} = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$- \frac{24 z}{11} = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Set the numerator equal to zero.

$24 z = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Divide each term in $24 z = 0$ by $24$ and simplify.

Divide each term in $24 z = 0$ by $24$ .

$\frac{24 z}{24} = \frac{0}{24}$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Simplify the left side.

Cancel the common factor of $24$ .

Cancel the common factor.

$\frac{24 z}{24} = \frac{0}{24}$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Divide $z$ by $1$ .

$z = \frac{0}{24}$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$z = \frac{0}{24}$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$z = \frac{0}{24}$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Simplify the right side.

Divide $0$ by $24$ .

$z = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$z = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$z = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

$z = 0$

$x = - 4 + \frac{12 z}{11}$

$y = - 1 + \frac{8 z}{11}$

Step 6

Replace all occurrences of $z$ with $0$ in each equation.

Replace all occurrences of $z$ in $x = - 4 + \frac{12 z}{11}$ with $0$ .

$x = - 4 + \frac{12 (0)}{11}$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

Simplify the right side.

Simplify $- 4 + \frac{12 (0)}{11}$ .

Multiply $12$ by $0$ .

$x = - 4 + \frac{0}{11}$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

Divide $0$ by $11$ .

$x = - 4 + 0$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

Add $- 4$ and $0$ .

$x = - 4$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

$x = - 4$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

$x = - 4$

$z = 0$

$y = - 1 + \frac{8 z}{11}$

Replace all occurrences of $z$ in $y = - 1 + \frac{8 z}{11}$ with $0$ .

$y = - 1 + \frac{8 (0)}{11}$

$x = - 4$

$z = 0$

Simplify the right side.

Simplify $- 1 + \frac{8 (0)}{11}$ .

Multiply $8$ by $0$ .

$y = - 1 + \frac{0}{11}$

$x = - 4$

$z = 0$

Divide $0$ by $11$ .

$y = - 1 + 0$

$x = - 4$

$z = 0$

Add $- 1$ and $0$ .

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

$y = - 1$

$x = - 4$

$z = 0$

Step 7

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

$(- 4, - 1, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 4, - 1, 0)$

Equation Form:

$x = - 4, y = - 1, z = 0$

Solve the System of Equations -5x+10y-4z=10 x+4y-4z=-8 x-7y+4z=3

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