Finite Math Examples

$- 5 x + y - 2 z = 52$ , $4 x - 4 y + 3 z = - 56$ , $- 2 x - 2 y - z = 20$

Step 1

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

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Step 1.1

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

$y - 2 z = 52 + 5 x$

$4 x - 4 y + 3 z = - 56$

$- 2 x - 2 y - z = 20$

Step 1.2

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

$y = 52 + 5 x + 2 z$

$4 x - 4 y + 3 z = - 56$

$- 2 x - 2 y - z = 20$

$y = 52 + 5 x + 2 z$

$4 x - 4 y + 3 z = - 56$

$- 2 x - 2 y - z = 20$

Step 2

Replace all occurrences of $y$ with $52 + 5 x + 2 z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $4 x - 4 y + 3 z = - 56$ with $52 + 5 x + 2 z$ .

$4 x - 4 (52 + 5 x + 2 z) + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 x - 4 (52 + 5 x + 2 z) + 3 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$4 x - 4 \cdot 52 - 4 (5 x) - 4 (2 z) + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 4$ by $52$ .

$4 x - 208 - 4 (5 x) - 4 (2 z) + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2.1.1.2.2

Multiply $5$ by $- 4$ .

$4 x - 208 - 20 x - 4 (2 z) + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2.1.1.2.3

Multiply $2$ by $- 4$ .

$4 x - 208 - 20 x - 8 z + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

$4 x - 208 - 20 x - 8 z + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

$4 x - 208 - 20 x - 8 z + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $20 x$ from $4 x$ .

$- 16 x - 208 - 8 z + 3 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.2.1.2.2

Add $- 8 z$ and $3 z$ .

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 2 y - z = 20$

Step 2.3

Replace all occurrences of $y$ in $- 2 x - 2 y - z = 20$ with $52 + 5 x + 2 z$ .

$- 2 x - 2 (52 + 5 x + 2 z) - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 2 x - 2 (52 + 5 x + 2 z) - z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$- 2 x - 2 \cdot 52 - 2 (5 x) - 2 (2 z) - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $52$ .

$- 2 x - 104 - 2 (5 x) - 2 (2 z) - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4.1.1.2.2

Multiply $5$ by $- 2$ .

$- 2 x - 104 - 10 x - 2 (2 z) - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4.1.1.2.3

Multiply $2$ by $- 2$ .

$- 2 x - 104 - 10 x - 4 z - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 104 - 10 x - 4 z - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 2 x - 104 - 10 x - 4 z - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $10 x$ from $- 2 x$ .

$- 12 x - 104 - 4 z - z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 2.4.1.2.2

Subtract $z$ from $- 4 z$ .

$- 12 x - 104 - 5 z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 12 x - 104 - 5 z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 12 x - 104 - 5 z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 12 x - 104 - 5 z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 12 x - 104 - 5 z = 20$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3

Solve for $x$ in $- 12 x - 104 - 5 z = 20$ .

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Step 3.1

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

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Step 3.1.1

Add $104$ to both sides of the equation.

$- 12 x - 5 z = 20 + 104$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.1.2

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

$- 12 x = 20 + 104 + 5 z$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.1.3

Add $20$ and $104$ .

$- 12 x = 124 + 5 z$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$- 12 x = 124 + 5 z$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2

Divide each term in $- 12 x = 124 + 5 z$ by $- 12$ and simplify.

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Step 3.2.1

Divide each term in $- 12 x = 124 + 5 z$ by $- 12$ .

$\frac{- 12 x}{- 12} = \frac{124}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 12$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 12 x}{- 12} = \frac{124}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{124}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = \frac{124}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = \frac{124}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $124$ and $- 12$ .

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Step 3.2.3.1.1.1

Factor $4$ out of $124$ .

$x = \frac{4 (31)}{- 12} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $4$ out of $- 12$ .

$x = \frac{4 \cdot 31}{4 \cdot - 3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 31}{4 \cdot - 3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{31}{- 3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = \frac{31}{- 3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = \frac{31}{- 3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{31}{3} + \frac{5 z}{- 12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 3.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{31}{3} - \frac{5 z}{12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$- 16 x - 208 - 5 z = - 56$

$y = 52 + 5 x + 2 z$

Step 4

Replace all occurrences of $x$ with $- \frac{31}{3} - \frac{5 z}{12}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 16 x - 208 - 5 z = - 56$ with $- \frac{31}{3} - \frac{5 z}{12}$ .

$- 16 (- \frac{31}{3} - \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 16 (- \frac{31}{3} - \frac{5 z}{12}) - 208 - 5 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 16 (- \frac{31}{3}) - 16 (- \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.2

Multiply $- 16 (- \frac{31}{3})$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $- 16$ .

$16 (\frac{31}{3}) - 16 (- \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.2.2

Combine $16$ and $\frac{31}{3}$ .

$\frac{16 \cdot 31}{3} - 16 (- \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.2.3

Multiply $16$ by $31$ .

$\frac{496}{3} - 16 (- \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{496}{3} - 16 (- \frac{5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.3

Cancel the common factor of $4$ .

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Step 4.2.1.1.3.1

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

$\frac{496}{3} - 16 \frac{- 5 z}{12} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.3.2

Factor $4$ out of $- 16$ .

$\frac{496}{3} + 4 (- 4) (\frac{- 5 z}{12}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.3.3

Factor $4$ out of $12$ .

$\frac{496}{3} + 4 \cdot (- 4 \frac{- 5 z}{4 \cdot 3}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.3.4

Cancel the common factor.

$\frac{496}{3} + 4 \cdot (- 4 \frac{- 5 z}{4 \cdot 3}) - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.3.5

Rewrite the expression.

$\frac{496}{3} - 4 \frac{- 5 z}{3} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{496}{3} - 4 \frac{- 5 z}{3} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.4

Combine $- 4$ and $\frac{- 5 z}{3}$ .

$\frac{496}{3} + \frac{- 4 (- 5 z)}{3} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.1.5

Multiply $- 5$ by $- 4$ .

$\frac{496}{3} + \frac{20 z}{3} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{496}{3} + \frac{20 z}{3} - 208 - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.2

To write $- 208$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{20 z}{3} + \frac{496}{3} - 208 \cdot \frac{3}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.3

Combine $- 208$ and $\frac{3}{3}$ .

$\frac{20 z}{3} + \frac{496}{3} + \frac{- 208 \cdot 3}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{20 z}{3} + \frac{496 - 208 \cdot 3}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 208$ by $3$ .

$\frac{20 z}{3} + \frac{496 - 624}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.5.2

Subtract $624$ from $496$ .

$\frac{20 z}{3} + \frac{- 128}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{20 z}{3} + \frac{- 128}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.6

Move the negative in front of the fraction.

$\frac{20 z}{3} - \frac{128}{3} - 5 z = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.7

To write $- 5 z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{20 z}{3} - 5 z \cdot \frac{3}{3} - \frac{128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.8

Combine $- 5 z$ and $\frac{3}{3}$ .

$\frac{20 z}{3} + \frac{- 5 z \cdot 3}{3} - \frac{128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{20 z - 5 z \cdot 3}{3} - \frac{128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.10

Combine the numerators over the common denominator.

$\frac{20 z - 5 z \cdot 3 - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.11

Multiply $3$ by $- 5$ .

$\frac{20 z - 15 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.2.1.12

Subtract $15 z$ from $20 z$ .

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + 5 x + 2 z$

Step 4.3

Replace all occurrences of $x$ in $y = 52 + 5 x + 2 z$ with $- \frac{31}{3} - \frac{5 z}{12}$ .

$y = 52 + 5 (- \frac{31}{3} - \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $52 + 5 (- \frac{31}{3} - \frac{5 z}{12}) + 2 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$y = 52 + 5 (- \frac{31}{3}) + 5 (- \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.2

Multiply $5 (- \frac{31}{3})$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $5$ .

$y = 52 - 5 (\frac{31}{3}) + 5 (- \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.2.2

Combine $- 5$ and $\frac{31}{3}$ .

$y = 52 + \frac{- 5 \cdot 31}{3} + 5 (- \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.2.3

Multiply $- 5$ by $31$ .

$y = 52 + \frac{- 155}{3} + 5 (- \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + \frac{- 155}{3} + 5 (- \frac{5 z}{12}) + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.3

Multiply $5 (- \frac{5 z}{12})$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $5$ .

$y = 52 + \frac{- 155}{3} - 5 \frac{5 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.3.2

Combine $- 5$ and $\frac{5 z}{12}$ .

$y = 52 + \frac{- 155}{3} + \frac{- 5 (5 z)}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.3.3

Multiply $5$ by $- 5$ .

$y = 52 + \frac{- 155}{3} + \frac{- 25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 + \frac{- 155}{3} + \frac{- 25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$y = 52 - \frac{155}{3} + \frac{- 25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.1.4.2

Move the negative in front of the fraction.

$y = 52 - \frac{155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 - \frac{155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 52 - \frac{155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.2

To write $52$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = 52 \cdot \frac{3}{3} - \frac{155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.3

Combine $52$ and $\frac{3}{3}$ .

$y = \frac{52 \cdot 3}{3} - \frac{155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{52 \cdot 3 - 155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $52$ by $3$ .

$y = \frac{156 - 155}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.5.2

Subtract $155$ from $156$ .

$y = \frac{1}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{25 z}{12} + 2 z$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.6

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$y = \frac{1}{3} - \frac{25 z}{12} + 2 z \cdot \frac{12}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.7

Simplify terms.

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Step 4.4.1.7.1

Combine $2 z$ and $\frac{12}{12}$ .

$y = \frac{1}{3} - \frac{25 z}{12} + \frac{2 z \cdot 12}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.7.2

Combine the numerators over the common denominator.

$y = \frac{1}{3} + \frac{- 25 z + 2 z \cdot 12}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} + \frac{- 25 z + 2 z \cdot 12}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8

Simplify each term.

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Step 4.4.1.8.1

Simplify the numerator.

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Step 4.4.1.8.1.1

Factor $z$ out of $- 25 z + 2 z \cdot 12$ .

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Step 4.4.1.8.1.1.1

Factor $z$ out of $- 25 z$ .

$y = \frac{1}{3} + \frac{z \cdot - 25 + 2 z \cdot 12}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.1.1.2

Factor $z$ out of $2 z \cdot 12$ .

$y = \frac{1}{3} + \frac{z \cdot - 25 + z (2 \cdot 12)}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.1.1.3

Factor $z$ out of $z \cdot - 25 + z (2 \cdot 12)$ .

$y = \frac{1}{3} + \frac{z (- 25 + 2 \cdot 12)}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} + \frac{z (- 25 + 2 \cdot 12)}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.1.2

Multiply $2$ by $12$ .

$y = \frac{1}{3} + \frac{z (- 25 + 24)}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.1.3

Add $- 25$ and $24$ .

$y = \frac{1}{3} + \frac{z \cdot - 1}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} + \frac{z \cdot - 1}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.2

Move $- 1$ to the left of $z$ .

$y = \frac{1}{3} + \frac{- 1 \cdot z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 4.4.1.8.3

Move the negative in front of the fraction.

$y = \frac{1}{3} - \frac{z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{z}{12}$

$\frac{5 z - 128}{3} = - 56$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5

Solve for $z$ in $\frac{5 z - 128}{3} = - 56$ .

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Step 5.1

Multiply both sides by $3$ .

$\frac{5 z - 128}{3} \cdot 3 = - 56 \cdot 3$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{5 z - 128}{3} \cdot 3 = - 56 \cdot 3$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.2.1.1.2

Rewrite the expression.

$5 z - 128 = - 56 \cdot 3$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$5 z - 128 = - 56 \cdot 3$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$5 z - 128 = - 56 \cdot 3$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 56$ by $3$ .

$5 z - 128 = - 168$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$5 z - 128 = - 168$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$5 z - 128 = - 168$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3

Solve for $z$ .

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Step 5.3.1

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

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Step 5.3.1.1

Add $128$ to both sides of the equation.

$5 z = - 168 + 128$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3.1.2

Add $- 168$ and $128$ .

$5 z = - 40$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$5 z = - 40$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3.2

Divide each term in $5 z = - 40$ by $5$ and simplify.

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Step 5.3.2.1

Divide each term in $5 z = - 40$ by $5$ .

$\frac{5 z}{5} = \frac{- 40}{5}$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $5$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{5 z}{5} = \frac{- 40}{5}$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 40}{5}$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = \frac{- 40}{5}$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = \frac{- 40}{5}$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 40$ by $5$ .

$z = - 8$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = - 8$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = - 8$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = - 8$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$z = - 8$

$y = \frac{1}{3} - \frac{z}{12}$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6

Replace all occurrences of $z$ with $- 8$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = \frac{1}{3} - \frac{z}{12}$ with $- 8$ .

$y = \frac{1}{3} - \frac{- 8}{12}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{1}{3} - \frac{- 8}{12}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Cancel the common factor of $- 8$ and $12$ .

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Step 6.2.1.1.1.1

Factor $4$ out of $- 8$ .

$y = \frac{1}{3} - \frac{4 (- 2)}{12}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.1.2

Cancel the common factors.

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Step 6.2.1.1.1.2.1

Factor $4$ out of $12$ .

$y = \frac{1}{3} - \frac{4 \cdot - 2}{4 \cdot 3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.1.2.2

Cancel the common factor.

$y = \frac{1}{3} - \frac{4 \cdot - 2}{4 \cdot 3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.1.2.3

Rewrite the expression.

$y = \frac{1}{3} - \frac{- 2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{- 2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} - \frac{- 2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.2

Move the negative in front of the fraction.

$y = \frac{1}{3} + \frac{2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.3

Multiply $- - \frac{2}{3}$ .

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Step 6.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{1}{3} + 1 (\frac{2}{3})$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.1.3.2

Multiply $\frac{2}{3}$ by $1$ .

$y = \frac{1}{3} + \frac{2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} + \frac{2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = \frac{1}{3} + \frac{2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.2

Combine fractions.

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Step 6.2.1.2.1

Combine the numerators over the common denominator.

$y = \frac{1 + 2}{3}$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.2.2

Simplify the expression.

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Step 6.2.1.2.2.1

Add $1$ and $2$ .

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

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.2.1.2.2.2

Divide $3$ by $3$ .

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 z}{12}$

Step 6.3

Replace all occurrences of $z$ in $x = - \frac{31}{3} - \frac{5 z}{12}$ with $- 8$ .

$x = - \frac{31}{3} - \frac{5 (- 8)}{12}$

$y = 1$

$z = - 8$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{31}{3} - \frac{5 (- 8)}{12}$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Cancel the common factor of $- 8$ and $12$ .

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Step 6.4.1.1.1.1

Factor $4$ out of $5 (- 8)$ .

$x = - \frac{31}{3} - \frac{4 (5 \cdot (- 2))}{12}$

$y = 1$

$z = - 8$

Step 6.4.1.1.1.2

Cancel the common factors.

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Step 6.4.1.1.1.2.1

Factor $4$ out of $12$ .

$x = - \frac{31}{3} - \frac{4 (5 \cdot (- 2))}{4 (3)}$

$y = 1$

$z = - 8$

Step 6.4.1.1.1.2.2

Cancel the common factor.

$x = - \frac{31}{3} - \frac{4 (5 \cdot (- 2))}{4 \cdot 3}$

$y = 1$

$z = - 8$

Step 6.4.1.1.1.2.3

Rewrite the expression.

$x = - \frac{31}{3} - \frac{5 \cdot (- 2)}{3}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 \cdot (- 2)}{3}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} - \frac{5 \cdot (- 2)}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.1.2

Multiply $5$ by $- 2$ .

$x = - \frac{31}{3} - \frac{- 10}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.1.3

Move the negative in front of the fraction.

$x = - \frac{31}{3} + \frac{10}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.1.4

Multiply $- - \frac{10}{3}$ .

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Step 6.4.1.1.4.1

Multiply $- 1$ by $- 1$ .

$x = - \frac{31}{3} + 1 (\frac{10}{3})$

$y = 1$

$z = - 8$

Step 6.4.1.1.4.2

Multiply $\frac{10}{3}$ by $1$ .

$x = - \frac{31}{3} + \frac{10}{3}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} + \frac{10}{3}$

$y = 1$

$z = - 8$

$x = - \frac{31}{3} + \frac{10}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.2

Combine fractions.

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Step 6.4.1.2.1

Combine the numerators over the common denominator.

$x = \frac{- 31 + 10}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.2.2

Simplify the expression.

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Step 6.4.1.2.2.1

Add $- 31$ and $10$ .

$x = \frac{- 21}{3}$

$y = 1$

$z = - 8$

Step 6.4.1.2.2.2

Divide $- 21$ by $3$ .

$x = - 7$

$y = 1$

$z = - 8$

$x = - 7$

$y = 1$

$z = - 8$

$x = - 7$

$y = 1$

$z = - 8$

$x = - 7$

$y = 1$

$z = - 8$

$x = - 7$

$y = 1$

$z = - 8$

$x = - 7$

$y = 1$

$z = - 8$

Step 7

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

$(- 7, 1, - 8)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 7, 1, - 8)$

Equation Form:

$x = - 7, y = 1, z = - 8$