Solve the System of Equations -5x+2y-5z=-4 y-5z=-7 -4x+2y+5z=5

$- 5 x + 2 y - 5 z = - 4$ $y - 5 z = - 7$ $- 4 x + 2 y + 5 z = 5$

Step 1

Add $5 z$ to both sides of the equation.

$y = - 7 + 5 z$

$- 5 x + 2 y - 5 z = - 4$

$- 4 x + 2 y + 5 z = 5$

Step 2

Replace all occurrences of $y$ with $- 7 + 5 z$ in each equation.

Replace all occurrences of $y$ in $- 5 x + 2 y - 5 z = - 4$ with $- 7 + 5 z$ .

$- 5 x + 2 (- 7 + 5 z) - 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

Simplify the left side.

Simplify $- 5 x + 2 (- 7 + 5 z) - 5 z$ .

Simplify each term.

Apply the distributive property.

$- 5 x + 2 \cdot - 7 + 2 (5 z) - 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

Multiply $2$ by $- 7$ .

$- 5 x - 14 + 2 (5 z) - 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

Multiply $5$ by $2$ .

$- 5 x - 14 + 10 z - 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

$- 5 x - 14 + 10 z - 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

Subtract $5 z$ from $10 z$ .

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x + 2 y + 5 z = 5$

Replace all occurrences of $y$ in $- 4 x + 2 y + 5 z = 5$ with $- 7 + 5 z$ .

$- 4 x + 2 (- 7 + 5 z) + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

Simplify the left side.

Simplify $- 4 x + 2 (- 7 + 5 z) + 5 z$ .

Simplify each term.

Apply the distributive property.

$- 4 x + 2 \cdot - 7 + 2 (5 z) + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

Multiply $2$ by $- 7$ .

$- 4 x - 14 + 2 (5 z) + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

Multiply $5$ by $2$ .

$- 4 x - 14 + 10 z + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x - 14 + 10 z + 5 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

Add $10 z$ and $5 z$ .

$- 4 x - 14 + 15 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x - 14 + 15 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x - 14 + 15 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

$- 4 x - 14 + 15 z = 5$

$- 5 x - 14 + 5 z = - 4$

$y = - 7 + 5 z$

Step 3

Reorder $- 7$ and $5 z$ .

$y = 5 z - 7$

$- 4 x - 14 + 15 z = 5$

$- 5 x - 14 + 5 z = - 4$

Step 4

Solve for $x$ in $- 4 x - 14 + 15 z = 5$ .

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

Add $14$ to both sides of the equation.

$- 4 x + 15 z = 5 + 14$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

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

$- 4 x = 5 + 14 - 15 z$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Add $5$ and $14$ .

$- 4 x = 19 - 15 z$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$- 4 x = 19 - 15 z$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Divide each term in $- 4 x = 19 - 15 z$ by $- 4$ and simplify.

Divide each term in $- 4 x = 19 - 15 z$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{19}{- 4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{19}{- 4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Divide $x$ by $1$ .

$x = \frac{19}{- 4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = \frac{19}{- 4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = \frac{19}{- 4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{19}{4} + \frac{- 15 z}{- 4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Dividing two negative values results in a positive value.

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- 5 x - 14 + 5 z = - 4$

Step 5

Replace all occurrences of $x$ with $- \frac{19}{4} + \frac{15 z}{4}$ in each equation.

Replace all occurrences of $x$ in $- 5 x - 14 + 5 z = - 4$ with $- \frac{19}{4} + \frac{15 z}{4}$ .

$- 5 (- \frac{19}{4} + \frac{15 z}{4}) - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the left side.

Simplify $- 5 (- \frac{19}{4} + \frac{15 z}{4}) - 14 + 5 z$ .

Simplify each term.

Apply the distributive property.

$- 5 (- \frac{19}{4}) - 5 \frac{15 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $- 5 (- \frac{19}{4})$ .

Multiply $- 1$ by $- 5$ .

$5 (\frac{19}{4}) - 5 \frac{15 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Combine $5$ and $\frac{19}{4}$ .

$\frac{5 \cdot 19}{4} - 5 \frac{15 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $5$ by $19$ .

$\frac{95}{4} - 5 \frac{15 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$\frac{95}{4} - 5 \frac{15 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $- 5 \frac{15 z}{4}$ .

Combine $- 5$ and $\frac{15 z}{4}$ .

$\frac{95}{4} + \frac{- 5 (15 z)}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $15$ by $- 5$ .

$\frac{95}{4} + \frac{- 75 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$\frac{95}{4} + \frac{- 75 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Move the negative in front of the fraction.

$\frac{95}{4} - \frac{75 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$\frac{95}{4} - \frac{75 z}{4} - 14 + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

To write $- 14$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{75 z}{4} + \frac{95}{4} - 14 \cdot \frac{4}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

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

$- \frac{75 z}{4} + \frac{95}{4} + \frac{- 14 \cdot 4}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Combine the numerators over the common denominator.

$- \frac{75 z}{4} + \frac{95 - 14 \cdot 4}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the numerator.

Multiply $- 14$ by $4$ .

$- \frac{75 z}{4} + \frac{95 - 56}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Subtract $56$ from $95$ .

$- \frac{75 z}{4} + \frac{39}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- \frac{75 z}{4} + \frac{39}{4} + 5 z = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

To write $5 z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{75 z}{4} + 5 z \cdot \frac{4}{4} + \frac{39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Combine $5 z$ and $\frac{4}{4}$ .

$- \frac{75 z}{4} + \frac{5 z \cdot 4}{4} + \frac{39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Combine the numerators over the common denominator.

$\frac{- 75 z + 5 z \cdot 4}{4} + \frac{39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Combine the numerators over the common denominator.

$\frac{- 75 z + 5 z \cdot 4 + 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $4$ by $5$ .

$\frac{- 75 z + 20 z + 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Add $- 75 z$ and $20 z$ .

$\frac{- 55 z + 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Factor $- 1$ out of $- 55 z$ .

$\frac{- (55 z) + 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Rewrite $39$ as $- 1 (- 39)$ .

$\frac{- (55 z) - 1 \cdot - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Factor $- 1$ out of $- (55 z) - 1 (- 39)$ .

$\frac{- (55 z - 39)}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the expression.

Rewrite $- (55 z - 39)$ as $- 1 (55 z - 39)$ .

$\frac{- 1 (55 z - 39)}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Move the negative in front of the fraction.

$- \frac{55 z - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- \frac{55 z - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- \frac{55 z - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- \frac{55 z - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$- \frac{55 z - 39}{4} = - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Step 6

Solve for $z$ in $- \frac{55 z - 39}{4} = - 4$ .

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{55 z - 39}{4}) = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 4 (- \frac{55 z - 39}{4})$ .

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{55 z - 39}{4}$ into the numerator.

$- 4 \frac{- (55 z - 39)}{4} = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{- (55 z - 39)}{4}) = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Cancel the common factor.

$4 \cdot (- 1 \frac{- (55 z - 39)}{4}) = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Rewrite the expression.

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (55 z - 39) = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Multiply $55 z - 39$ by $1$ .

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = - 4 \cdot - 4$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the right side.

Multiply $- 4$ by $- 4$ .

$55 z - 39 = 16$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = 16$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z - 39 = 16$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

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

Add $39$ to both sides of the equation.

$55 z = 16 + 39$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Add $16$ and $39$ .

$55 z = 55$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$55 z = 55$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Divide each term in $55 z = 55$ by $55$ and simplify.

Divide each term in $55 z = 55$ by $55$ .

$\frac{55 z}{55} = \frac{55}{55}$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the left side.

Cancel the common factor of $55$ .

Cancel the common factor.

$\frac{55 z}{55} = \frac{55}{55}$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Divide $z$ by $1$ .

$z = \frac{55}{55}$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$z = \frac{55}{55}$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$z = \frac{55}{55}$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Simplify the right side.

Divide $55$ by $55$ .

$z = 1$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$z = 1$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$z = 1$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

$z = 1$

$x = - \frac{19}{4} + \frac{15 z}{4}$

$y = 5 z - 7$

Step 7

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

Replace all occurrences of $z$ in $x = - \frac{19}{4} + \frac{15 z}{4}$ with $1$ .

$x = - \frac{19}{4} + \frac{15 (1)}{4}$

$z = 1$

$y = 5 z - 7$

Simplify the right side.

Simplify $- \frac{19}{4} + \frac{15 (1)}{4}$ .

Combine the numerators over the common denominator.

$x = \frac{- 19 + 15 (1)}{4}$

$z = 1$

$y = 5 z - 7$

Simplify the expression.

Multiply $15$ by $1$ .

$x = \frac{- 19 + 15}{4}$

$z = 1$

$y = 5 z - 7$

Add $- 19$ and $15$ .

$x = \frac{- 4}{4}$

$z = 1$

$y = 5 z - 7$

Divide $- 4$ by $4$ .

$x = - 1$

$z = 1$

$y = 5 z - 7$

$x = - 1$

$z = 1$

$y = 5 z - 7$

$x = - 1$

$z = 1$

$y = 5 z - 7$

$x = - 1$

$z = 1$

$y = 5 z - 7$

Replace all occurrences of $z$ in $y = 5 z - 7$ with $1$ .

$y = 5 (1) - 7$

$x = - 1$

$z = 1$

Simplify the right side.

Simplify $5 (1) - 7$ .

Multiply $5$ by $1$ .

$y = 5 - 7$

$x = - 1$

$z = 1$

Subtract $7$ from $5$ .

$y = - 2$

$x = - 1$

$z = 1$

$y = - 2$

$x = - 1$

$z = 1$

$y = - 2$

$x = - 1$

$z = 1$

$y = - 2$

$x = - 1$

$z = 1$

Step 8

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

$(- 1, - 2, 1)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(- 1, - 2, 1)$

Equation Form:

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

Solve the System of Equations -5x+2y-5z=-4 y-5z=-7 -4x+2y+5z=5

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