Algebra Examples

$2 x - 3 y + 10 z = - 83$ $12 y + 5 z = - 23$ $- 3 y + 2 z = - 17$

Step 1

Solve for $x$ in $2 x - 3 y + 10 z = - 83$ .

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Step 1.1

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

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Step 1.1.1

Add $3 y$ to both sides of the equation.

$2 x + 10 z = - 83 + 3 y$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.1.2

Subtract $10 z$ from both sides of the equation.

$2 x = - 83 + 3 y - 10 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$2 x = - 83 + 3 y - 10 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2

Divide each term in $2 x = - 83 + 3 y - 10 z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = - 83 + 3 y - 10 z$ by $2$ .

$\frac{2 x}{2} = \frac{- 83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = \frac{- 83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = \frac{- 83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{83}{2} + \frac{3 y}{2} + \frac{- 10 z}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3.1.2

Cancel the common factor of $- 10$ and $2$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $- 10 z$ .

$x = - \frac{83}{2} + \frac{3 y}{2} + \frac{2 (- 5 z)}{2}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $2$ .

$x = - \frac{83}{2} + \frac{3 y}{2} + \frac{2 (- 5 z)}{2 (1)}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{83}{2} + \frac{3 y}{2} + \frac{2 (- 5 z)}{2 \cdot 1}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{83}{2} + \frac{3 y}{2} + \frac{- 5 z}{1}$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 1.2.3.1.2.2.4

Divide $- 5 z$ by $1$ .

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$12 y + 5 z = - 23$

$- 3 y + 2 z = - 17$

Step 2

Solve for $y$ in $12 y + 5 z = - 23$ .

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Step 2.1

Subtract $5 z$ from both sides of the equation.

$12 y = - 23 - 5 z$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 2.2

Divide each term in $12 y = - 23 - 5 z$ by $12$ and simplify.

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Step 2.2.1

Divide each term in $12 y = - 23 - 5 z$ by $12$ .

$\frac{12 y}{12} = \frac{- 23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $12$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{- 23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = \frac{- 23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = \frac{- 23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{23}{12} + \frac{- 5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$

$- 3 y + 2 z = - 17$

Step 3

Replace all occurrences of $y$ with $- \frac{23}{12} - \frac{5 z}{12}$ in each equation.

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Step 3.1

Replace all occurrences of $y$ in $x = - \frac{83}{2} + \frac{3 y}{2} - 5 z$ with $- \frac{23}{12} - \frac{5 z}{12}$ .

$x = - \frac{83}{2} + \frac{3 (- \frac{23}{12} - \frac{5 z}{12})}{2} - 5 z$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{83}{2} + \frac{3 (- \frac{23}{12} - \frac{5 z}{12})}{2} - 5 z$ .

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Step 3.2.1.1

Combine the numerators over the common denominator.

$x = - 5 z + \frac{- 83 + 3 (- \frac{23}{12} - \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

Apply the distributive property.

$x = - 5 z + \frac{- 83 + 3 (- \frac{23}{12}) + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.2.1

Move the leading negative in $- \frac{23}{12}$ into the numerator.

$x = - 5 z + \frac{- 83 + 3 (\frac{- 23}{12}) + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.2.2

Factor $3$ out of $12$ .

$x = - 5 z + \frac{- 83 + 3 (\frac{- 23}{3 (4)}) + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.2.3

Cancel the common factor.

$x = - 5 z + \frac{- 83 + 3 (\frac{- 23}{3 \cdot 4}) + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.2.4

Rewrite the expression.

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + 3 (- \frac{5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.3

Cancel the common factor of $3$ .

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Step 3.2.1.2.3.1

Move the leading negative in $- \frac{5 z}{12}$ into the numerator.

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + 3 (\frac{- 5 z}{12})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.3.2

Factor $3$ out of $12$ .

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + 3 (\frac{- 5 z}{3 (4)})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.3.3

Cancel the common factor.

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + 3 (\frac{- 5 z}{3 \cdot 4})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.3.4

Rewrite the expression.

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + \frac{- 5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- 83 + \frac{- 23}{4} + \frac{- 5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.4

Simplify each term.

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Step 3.2.1.2.4.1

Move the negative in front of the fraction.

$x = - 5 z + \frac{- 83 - \frac{23}{4} + \frac{- 5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.2.4.2

Move the negative in front of the fraction.

$x = - 5 z + \frac{- 83 - \frac{23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- 83 - \frac{23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- 83 - \frac{23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.3

To write $- 83$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 5 z + \frac{- 83 \cdot \frac{4}{4} - \frac{23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.4

Combine $- 83$ and $\frac{4}{4}$ .

$x = - 5 z + \frac{\frac{- 83 \cdot 4}{4} - \frac{23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.5

Combine the numerators over the common denominator.

$x = - 5 z + \frac{\frac{- 83 \cdot 4 - 23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.6

Simplify the numerator.

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Step 3.2.1.6.1

Multiply $- 83$ by $4$ .

$x = - 5 z + \frac{\frac{- 332 - 23}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.6.2

Subtract $23$ from $- 332$ .

$x = - 5 z + \frac{\frac{- 355}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{\frac{- 355}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.7

Move the negative in front of the fraction.

$x = - 5 z + \frac{- \frac{355}{4} - \frac{5 z}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8

Simplify each term.

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Step 3.2.1.8.1

Simplify the numerator.

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Step 3.2.1.8.1.1

Factor $- 1$ out of $- \frac{355}{4} - \frac{5 z}{4}$ .

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Step 3.2.1.8.1.1.1

Reorder $- \frac{355}{4}$ and $- \frac{5 z}{4}$ .

$x = - 5 z + \frac{- \frac{5 z}{4} - \frac{355}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.1.2

Factor $- 1$ out of $- \frac{5 z}{4}$ .

$x = - 5 z + \frac{- (\frac{5 z}{4}) - \frac{355}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.1.3

Factor $- 1$ out of $- \frac{355}{4}$ .

$x = - 5 z + \frac{- (\frac{5 z}{4}) - (\frac{355}{4})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.1.4

Factor $- 1$ out of $- (\frac{5 z}{4}) - (\frac{355}{4})$ .

$x = - 5 z + \frac{- (\frac{5 z}{4} + \frac{355}{4})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- (\frac{5 z}{4} + \frac{355}{4})}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.2

Combine the numerators over the common denominator.

$x = - 5 z + \frac{- \frac{5 z + 355}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.3

Factor $5$ out of $5 z + 355$ .

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Step 3.2.1.8.1.3.1

Factor $5$ out of $5 z$ .

$x = - 5 z + \frac{- \frac{5 (z) + 355}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.3.2

Factor $5$ out of $355$ .

$x = - 5 z + \frac{- \frac{5 z + 5 \cdot 71}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.1.3.3

Factor $5$ out of $5 z + 5 \cdot 71$ .

$x = - 5 z + \frac{- \frac{5 (z + 71)}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- \frac{5 (z + 71)}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z + \frac{- \frac{5 (z + 71)}{4}}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.2

Multiply the numerator by the reciprocal of the denominator.

$x = - 5 z - \frac{5 (z + 71)}{4} \cdot \frac{1}{2}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.3

Multiply $- \frac{5 (z + 71)}{4} \cdot \frac{1}{2}$ .

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Step 3.2.1.8.3.1

Multiply $\frac{1}{2}$ by $\frac{5 (z + 71)}{4}$ .

$x = - 5 z - \frac{5 (z + 71)}{2 \cdot 4}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.8.3.2

Multiply $2$ by $4$ .

$x = - 5 z - \frac{5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z - \frac{5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - 5 z - \frac{5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.9

To write $- 5 z$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x = - 5 z \cdot \frac{8}{8} - \frac{5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.10

Simplify terms.

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Step 3.2.1.10.1

Combine $- 5 z$ and $\frac{8}{8}$ .

$x = \frac{- 5 z \cdot 8}{8} - \frac{5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.10.2

Combine the numerators over the common denominator.

$x = \frac{- 5 z \cdot 8 - 5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = \frac{- 5 z \cdot 8 - 5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11

Simplify the numerator.

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Step 3.2.1.11.1

Factor $5$ out of $- 5 z \cdot 8 - 5 (z + 71)$ .

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Step 3.2.1.11.1.1

Factor $5$ out of $- 5 z \cdot 8$ .

$x = \frac{5 (- z \cdot 8) - 5 (z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.1.2

Factor $5$ out of $- 5 (z + 71)$ .

$x = \frac{5 (- z \cdot 8) + 5 (- (z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.1.3

Factor $5$ out of $5 (- z \cdot 8) + 5 (- (z + 71))$ .

$x = \frac{5 (- z \cdot 8 - (z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = \frac{5 (- z \cdot 8 - (z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.2

Multiply $8$ by $- 1$ .

$x = \frac{5 (- 8 z - (z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.3

Apply the distributive property.

$x = \frac{5 (- 8 z - z - 1 \cdot 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.4

Multiply $- 1$ by $71$ .

$x = \frac{5 (- 8 z - z - 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.11.5

Subtract $z$ from $- 8 z$ .

$x = \frac{5 (- 9 z - 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = \frac{5 (- 9 z - 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.12

Simplify with factoring out.

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Step 3.2.1.12.1

Factor $- 1$ out of $- 9 z$ .

$x = \frac{5 (- (9 z) - 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.12.2

Rewrite $- 71$ as $- 1 (71)$ .

$x = \frac{5 (- (9 z) - 1 \cdot 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.12.3

Factor $- 1$ out of $- (9 z) - 1 (71)$ .

$x = \frac{5 (- (9 z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.12.4

Simplify the expression.

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Step 3.2.1.12.4.1

Rewrite $- (9 z + 71)$ as $- 1 (9 z + 71)$ .

$x = \frac{5 (- 1 (9 z + 71))}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.2.1.12.4.2

Move the negative in front of the fraction.

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 3 y + 2 z = - 17$

Step 3.3

Replace all occurrences of $y$ in $- 3 y + 2 z = - 17$ with $- \frac{23}{12} - \frac{5 z}{12}$ .

$- 3 (- \frac{23}{12} - \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify $- 3 (- \frac{23}{12} - \frac{5 z}{12}) + 2 z$ .

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Apply the distributive property.

$- 3 (- \frac{23}{12}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.2.1

Move the leading negative in $- \frac{23}{12}$ into the numerator.

$- 3 (\frac{- 23}{12}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.2.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- 23}{12}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.2.3

Factor $3$ out of $12$ .

$3 \cdot (- 1 \frac{- 23}{3 \cdot 4}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.2.4

Cancel the common factor.

$3 \cdot (- 1 \frac{- 23}{3 \cdot 4}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.2.5

Rewrite the expression.

$- 1 (\frac{- 23}{4}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 1 (\frac{- 23}{4}) - 3 (- \frac{5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.3

Cancel the common factor of $3$ .

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Step 3.4.1.1.3.1

Move the leading negative in $- \frac{5 z}{12}$ into the numerator.

$- 1 (\frac{- 23}{4}) - 3 \frac{- 5 z}{12} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.3.2

Factor $3$ out of $- 3$ .

$- 1 (\frac{- 23}{4}) + 3 (- 1) (\frac{- 5 z}{12}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.3.3

Factor $3$ out of $12$ .

$- 1 (\frac{- 23}{4}) + 3 \cdot (- 1 \frac{- 5 z}{3 \cdot 4}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.3.4

Cancel the common factor.

$- 1 (\frac{- 23}{4}) + 3 \cdot (- 1 \frac{- 5 z}{3 \cdot 4}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.3.5

Rewrite the expression.

$- 1 (\frac{- 23}{4}) - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$- 1 (\frac{- 23}{4}) - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4

Simplify each term.

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Step 3.4.1.1.4.1

Move the negative in front of the fraction.

$- 1 (- \frac{23}{4}) - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4.2

Multiply $- 1 (- \frac{23}{4})$ .

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Step 3.4.1.1.4.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{23}{4}) - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4.2.2

Multiply $\frac{23}{4}$ by $1$ .

$\frac{23}{4} - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23}{4} - 1 \frac{- 5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4.3

Move the negative in front of the fraction.

$\frac{23}{4} - 1 (- \frac{5 z}{4}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4.4

Multiply $- 1 (- \frac{5 z}{4})$ .

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Step 3.4.1.1.4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{23}{4} + 1 (\frac{5 z}{4}) + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.1.4.4.2

Multiply $\frac{5 z}{4}$ by $1$ .

$\frac{23}{4} + \frac{5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23}{4} + \frac{5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23}{4} + \frac{5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23}{4} + \frac{5 z}{4} + 2 z = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.2

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

$\frac{23}{4} + \frac{5 z}{4} + 2 z \cdot \frac{4}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.3

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

$\frac{23}{4} + \frac{5 z}{4} + \frac{2 z \cdot 4}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.4

Combine the numerators over the common denominator.

$\frac{23}{4} + \frac{5 z + 2 z \cdot 4}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.5

Combine the numerators over the common denominator.

$\frac{23 + 5 z + 2 z \cdot 4}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.6

Multiply $4$ by $2$ .

$\frac{23 + 5 z + 8 z}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 3.4.1.7

Add $5 z$ and $8 z$ .

$\frac{23 + 13 z}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23 + 13 z}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23 + 13 z}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$\frac{23 + 13 z}{4} = - 17$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4

Solve for $z$ in $\frac{23 + 13 z}{4} = - 17$ .

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Step 4.1

Multiply both sides by $4$ .

$\frac{23 + 13 z}{4} \cdot 4 = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $\frac{23 + 13 z}{4} \cdot 4$ .

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Step 4.2.1.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{23 + 13 z}{4} \cdot 4 = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.2.1.1.1.2

Rewrite the expression.

$23 + 13 z = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$23 + 13 z = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.2.1.1.2

Reorder $23$ and $13 z$ .

$13 z + 23 = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$13 z + 23 = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$13 z + 23 = - 17 \cdot 4$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 17$ by $4$ .

$13 z + 23 = - 68$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$13 z + 23 = - 68$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$13 z + 23 = - 68$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3

Solve for $z$ .

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Step 4.3.1

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

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Step 4.3.1.1

Subtract $23$ from both sides of the equation.

$13 z = - 68 - 23$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3.1.2

Subtract $23$ from $- 68$ .

$13 z = - 91$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$13 z = - 91$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3.2

Divide each term in $13 z = - 91$ by $13$ and simplify.

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Step 4.3.2.1

Divide each term in $13 z = - 91$ by $13$ .

$\frac{13 z}{13} = \frac{- 91}{13}$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $13$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{13 z}{13} = \frac{- 91}{13}$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 91}{13}$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = \frac{- 91}{13}$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = \frac{- 91}{13}$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $- 91$ by $13$ .

$z = - 7$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = - 7$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = - 7$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = - 7$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$z = - 7$

$x = - \frac{5 (9 z + 71)}{8}$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5

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

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Step 5.1

Replace all occurrences of $z$ in $x = - \frac{5 (9 z + 71)}{8}$ with $- 7$ .

$x = - \frac{5 (9 (- 7) + 71)}{8}$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{5 (9 (- 7) + 71)}{8}$ .

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Step 5.2.1.1

Simplify the numerator.

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Step 5.2.1.1.1

Multiply $9$ by $- 7$ .

$x = - \frac{5 (- 63 + 71)}{8}$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.2.1.1.2

Add $- 63$ and $71$ .

$x = - \frac{5 \cdot 8}{8}$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - \frac{5 \cdot 8}{8}$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.2.1.2

Simplify the expression.

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Step 5.2.1.2.1

Multiply $5$ by $8$ .

$x = - \frac{40}{8}$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.2.1.2.2

Divide $40$ by $8$ .

$x = - 1 \cdot 5$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.2.1.2.3

Multiply $- 1$ by $5$ .

$x = - 5$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - 5$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - 5$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

$x = - 5$

$z = - 7$

$y = - \frac{23}{12} - \frac{5 z}{12}$

Step 5.3

Replace all occurrences of $z$ in $y = - \frac{23}{12} - \frac{5 z}{12}$ with $- 7$ .

$y = - \frac{23}{12} - \frac{5 (- 7)}{12}$

$x = - 5$

$z = - 7$

Step 5.4

Simplify the right side.

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Step 5.4.1

Simplify $- \frac{23}{12} - \frac{5 (- 7)}{12}$ .

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Step 5.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 23 - 5 \cdot - 7}{12}$

$x = - 5$

$z = - 7$

Step 5.4.1.2

Simplify the expression.

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Step 5.4.1.2.1

Multiply $- 5$ by $- 7$ .

$y = \frac{- 23 + 35}{12}$

$x = - 5$

$z = - 7$

Step 5.4.1.2.2

Add $- 23$ and $35$ .

$y = \frac{12}{12}$

$x = - 5$

$z = - 7$

Step 5.4.1.2.3

Divide $12$ by $12$ .

$y = 1$

$x = - 5$

$z = - 7$

$y = 1$

$x = - 5$

$z = - 7$

$y = 1$

$x = - 5$

$z = - 7$

$y = 1$

$x = - 5$

$z = - 7$

$y = 1$

$x = - 5$

$z = - 7$

Step 6

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

$(- 5, 1, - 7)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 5, 1, - 7)$

Equation Form:

$x = - 5, y = 1, z = - 7$