Precalculus Examples

$32 x - 9 y + 37 z = - 17$ , $41 x + 8 y - 6 z = 40$ , $- 13 x + 19 y + 40 z = - 6$

Step 1

Solve for $x$ in $32 x - 9 y + 37 z = - 17$ .

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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 $9 y$ to both sides of the equation.

$32 x + 37 z = - 17 + 9 y$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 1.1.2

Subtract $37 z$ from both sides of the equation.

$32 x = - 17 + 9 y - 37 z$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$32 x = - 17 + 9 y - 37 z$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 1.2

Divide each term in $32 x = - 17 + 9 y - 37 z$ by $32$ and simplify.

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Step 1.2.1

Divide each term in $32 x = - 17 + 9 y - 37 z$ by $32$ .

$\frac{32 x}{32} = \frac{- 17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $32$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{32 x}{32} = \frac{- 17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = \frac{- 17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = \frac{- 17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

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.

$x = - \frac{17}{32} + \frac{9 y}{32} + \frac{- 37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$41 x + 8 y - 6 z = 40$

$- 13 x + 19 y + 40 z = - 6$

Step 2

Replace all occurrences of $x$ with $- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $41 x + 8 y - 6 z = 40$ with $- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$ .

$41 (- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $41 (- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}) + 8 y - 6 z$ .

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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.

$41 (- \frac{17}{32}) + 41 (\frac{9 y}{32}) + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $41 (- \frac{17}{32})$ .

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Step 2.2.1.1.2.1.1

Multiply $- 1$ by $41$ .

$- 41 (\frac{17}{32}) + 41 (\frac{9 y}{32}) + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.1.2

Combine $- 41$ and $\frac{17}{32}$ .

$\frac{- 41 \cdot 17}{32} + 41 (\frac{9 y}{32}) + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.1.3

Multiply $- 41$ by $17$ .

$\frac{- 697}{32} + 41 (\frac{9 y}{32}) + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 697}{32} + 41 (\frac{9 y}{32}) + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.2

Multiply $41 \frac{9 y}{32}$ .

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Step 2.2.1.1.2.2.1

Combine $41$ and $\frac{9 y}{32}$ .

$\frac{- 697}{32} + \frac{41 (9 y)}{32} + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.2.2

Multiply $9$ by $41$ .

$\frac{- 697}{32} + \frac{369 y}{32} + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 697}{32} + \frac{369 y}{32} + 41 (- \frac{37 z}{32}) + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.3

Multiply $41 (- \frac{37 z}{32})$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $41$ .

$\frac{- 697}{32} + \frac{369 y}{32} - 41 \frac{37 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.3.2

Combine $- 41$ and $\frac{37 z}{32}$ .

$\frac{- 697}{32} + \frac{369 y}{32} + \frac{- 41 (37 z)}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.2.3.3

Multiply $37$ by $- 41$ .

$\frac{- 697}{32} + \frac{369 y}{32} + \frac{- 1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 697}{32} + \frac{369 y}{32} + \frac{- 1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 697}{32} + \frac{369 y}{32} + \frac{- 1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.3

Simplify each term.

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Step 2.2.1.1.3.1

Move the negative in front of the fraction.

$- \frac{697}{32} + \frac{369 y}{32} + \frac{- 1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.1.3.2

Move the negative in front of the fraction.

$- \frac{697}{32} + \frac{369 y}{32} - \frac{1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{697}{32} + \frac{369 y}{32} - \frac{1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{697}{32} + \frac{369 y}{32} - \frac{1517 z}{32} + 8 y - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.2

To write $8 y$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$- \frac{697}{32} - \frac{1517 z}{32} + \frac{369 y}{32} + 8 y \cdot \frac{32}{32} - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.3

Combine $8 y$ and $\frac{32}{32}$ .

$- \frac{697}{32} - \frac{1517 z}{32} + \frac{369 y}{32} + \frac{8 y \cdot 32}{32} - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{697}{32} - \frac{1517 z}{32} + \frac{369 y + 8 y \cdot 32}{32} - 6 z = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 6 z + \frac{- 697 - 1517 z + 369 y + 8 y \cdot 32}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.6

Multiply $32$ by $8$ .

$- 6 z + \frac{- 697 - 1517 z + 369 y + 256 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.7

Add $369 y$ and $256 y$ .

$- 6 z + \frac{- 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.8

To write $- 6 z$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$- 6 z \cdot \frac{32}{32} + \frac{- 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

Combine $- 6 z$ and $\frac{32}{32}$ .

$\frac{- 6 z \cdot 32}{32} + \frac{- 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{- 6 z \cdot 32 - 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 6 z \cdot 32 - 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $32$ by $- 6$ .

$\frac{- 192 z - 697 - 1517 z + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.10.2

Subtract $1517 z$ from $- 192 z$ .

$\frac{- 1709 z - 697 + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$\frac{- 1709 z - 697 + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11

Simplify with factoring out.

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Step 2.2.1.11.1

Factor $- 1$ out of $- 1709 z$ .

$\frac{- (1709 z) - 697 + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.2

Rewrite $- 697$ as $- 1 (697)$ .

$\frac{- (1709 z) - 1 \cdot 697 + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.3

Factor $- 1$ out of $- (1709 z) - 1 (697)$ .

$\frac{- (1709 z + 697) + 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.4

Factor $- 1$ out of $625 y$ .

$\frac{- (1709 z + 697) - (- 625 y)}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.5

Factor $- 1$ out of $- (1709 z + 697) - (- 625 y)$ .

$\frac{- (1709 z + 697 - 625 y)}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.6

Simplify the expression.

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Step 2.2.1.11.6.1

Rewrite $- (1709 z + 697 - 625 y)$ as $- 1 (1709 z + 697 - 625 y)$ .

$\frac{- 1 (1709 z + 697 - 625 y)}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.2.1.11.6.2

Move the negative in front of the fraction.

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- 13 x + 19 y + 40 z = - 6$

Step 2.3

Replace all occurrences of $x$ in $- 13 x + 19 y + 40 z = - 6$ with $- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$ .

$- 13 (- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 13 (- \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}) + 19 y + 40 z$ .

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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.

$- 13 (- \frac{17}{32}) - 13 \frac{9 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 13 (- \frac{17}{32})$ .

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Step 2.4.1.1.2.1.1

Multiply $- 1$ by $- 13$ .

$13 (\frac{17}{32}) - 13 \frac{9 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.1.2

Combine $13$ and $\frac{17}{32}$ .

$\frac{13 \cdot 17}{32} - 13 \frac{9 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.1.3

Multiply $13$ by $17$ .

$\frac{221}{32} - 13 \frac{9 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{221}{32} - 13 \frac{9 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.2

Multiply $- 13 \frac{9 y}{32}$ .

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Step 2.4.1.1.2.2.1

Combine $- 13$ and $\frac{9 y}{32}$ .

$\frac{221}{32} + \frac{- 13 (9 y)}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.2.2

Multiply $9$ by $- 13$ .

$\frac{221}{32} + \frac{- 117 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{221}{32} + \frac{- 117 y}{32} - 13 (- \frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.3

Multiply $- 13 (- \frac{37 z}{32})$ .

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Step 2.4.1.1.2.3.1

Multiply $- 1$ by $- 13$ .

$\frac{221}{32} + \frac{- 117 y}{32} + 13 (\frac{37 z}{32}) + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.3.2

Combine $13$ and $\frac{37 z}{32}$ .

$\frac{221}{32} + \frac{- 117 y}{32} + \frac{13 (37 z)}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.2.3.3

Multiply $37$ by $13$ .

$\frac{221}{32} + \frac{- 117 y}{32} + \frac{481 z}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{221}{32} + \frac{- 117 y}{32} + \frac{481 z}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{221}{32} + \frac{- 117 y}{32} + \frac{481 z}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.1.3

Move the negative in front of the fraction.

$\frac{221}{32} - \frac{117 y}{32} + \frac{481 z}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{221}{32} - \frac{117 y}{32} + \frac{481 z}{32} + 19 y + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.2

To write $19 y$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$\frac{221}{32} + \frac{481 z}{32} - \frac{117 y}{32} + 19 y \cdot \frac{32}{32} + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.3

Combine $19 y$ and $\frac{32}{32}$ .

$\frac{221}{32} + \frac{481 z}{32} - \frac{117 y}{32} + \frac{19 y \cdot 32}{32} + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.4

Combine the numerators over the common denominator.

$\frac{221}{32} + \frac{481 z}{32} + \frac{- 117 y + 19 y \cdot 32}{32} + 40 z = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.5

Combine the numerators over the common denominator.

$40 z + \frac{221 + 481 z - 117 y + 19 y \cdot 32}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.6

Multiply $32$ by $19$ .

$40 z + \frac{221 + 481 z - 117 y + 608 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.7

Add $- 117 y$ and $608 y$ .

$40 z + \frac{221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.8

To write $40 z$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$40 z \cdot \frac{32}{32} + \frac{221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.9

Simplify terms.

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Step 2.4.1.9.1

Combine $40 z$ and $\frac{32}{32}$ .

$\frac{40 z \cdot 32}{32} + \frac{221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.9.2

Combine the numerators over the common denominator.

$\frac{40 z \cdot 32 + 221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{40 z \cdot 32 + 221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.10

Simplify the numerator.

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Step 2.4.1.10.1

Multiply $32$ by $40$ .

$\frac{1280 z + 221 + 481 z + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 2.4.1.10.2

Add $1280 z$ and $481 z$ .

$\frac{1761 z + 221 + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{1761 z + 221 + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{1761 z + 221 + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{1761 z + 221 + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{1761 z + 221 + 491 y}{32} = - 6$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3

Solve for $z$ in $\frac{1761 z + 221 + 491 y}{32} = - 6$ .

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Step 3.1

Multiply both sides by $32$ .

$\frac{1761 z + 221 + 491 y}{32} \cdot 32 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{1761 z + 221 + 491 y}{32} \cdot 32$ .

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Step 3.2.1.1.1

Cancel the common factor of $32$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{1761 z + 221 + 491 y}{32} \cdot 32 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.2.1.1.1.2

Rewrite the expression.

$1761 z + 221 + 491 y = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$1761 z + 221 + 491 y = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $221$ .

$1761 z + 491 y + 221 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.2.1.1.2.2

Reorder $1761 z$ and $491 y$ .

$491 y + 1761 z + 221 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$491 y + 1761 z + 221 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$491 y + 1761 z + 221 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$491 y + 1761 z + 221 = - 6 \cdot 32$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 6$ by $32$ .

$491 y + 1761 z + 221 = - 192$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$491 y + 1761 z + 221 = - 192$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$491 y + 1761 z + 221 = - 192$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3

Solve for $z$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $491 y$ from both sides of the equation.

$1761 z + 221 = - 192 - 491 y$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.1.2

Subtract $221$ from both sides of the equation.

$1761 z = - 192 - 491 y - 221$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.1.3

Subtract $221$ from $- 192$ .

$1761 z = - 491 y - 413$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$1761 z = - 491 y - 413$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.2

Divide each term in $1761 z = - 491 y - 413$ by $1761$ and simplify.

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Step 3.3.2.1

Divide each term in $1761 z = - 491 y - 413$ by $1761$ .

$\frac{1761 z}{1761} = \frac{- 491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $1761$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{1761 z}{1761} = \frac{- 491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = \frac{- 491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = \frac{- 491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Simplify each term.

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Step 3.3.2.3.1.1

Move the negative in front of the fraction.

$z = - \frac{491 y}{1761} + \frac{- 413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 3.3.2.3.1.2

Move the negative in front of the fraction.

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$- \frac{1709 z + 697 - 625 y}{32} = 40$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4

Replace all occurrences of $z$ with $- \frac{491 y}{1761} - \frac{413}{1761}$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- \frac{1709 z + 697 - 625 y}{32} = 40$ with $- \frac{491 y}{1761} - \frac{413}{1761}$ .

$- \frac{1709 (- \frac{491 y}{1761} - \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- \frac{1709 (- \frac{491 y}{1761} - \frac{413}{1761}) + 697 - 625 y}{32}$ .

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Step 4.2.1.1

Simplify the numerator.

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Step 4.2.1.1.1

Apply the distributive property.

$- \frac{1709 (- \frac{491 y}{1761}) + 1709 (- \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.2

Multiply $1709 (- \frac{491 y}{1761})$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $1709$ .

$- \frac{- 1709 \frac{491 y}{1761} + 1709 (- \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.2.2

Combine $- 1709$ and $\frac{491 y}{1761}$ .

$- \frac{\frac{- 1709 (491 y)}{1761} + 1709 (- \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.2.3

Multiply $491$ by $- 1709$ .

$- \frac{\frac{- 839119 y}{1761} + 1709 (- \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{\frac{- 839119 y}{1761} + 1709 (- \frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.3

Multiply $1709 (- \frac{413}{1761})$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $1709$ .

$- \frac{\frac{- 839119 y}{1761} - 1709 (\frac{413}{1761}) + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.3.2

Combine $- 1709$ and $\frac{413}{1761}$ .

$- \frac{\frac{- 839119 y}{1761} + \frac{- 1709 \cdot 413}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.3.3

Multiply $- 1709$ by $413$ .

$- \frac{\frac{- 839119 y}{1761} + \frac{- 705817}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{\frac{- 839119 y}{1761} + \frac{- 705817}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.4

Simplify each term.

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Step 4.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{- \frac{839119 y}{1761} + \frac{- 705817}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{- \frac{839119 y}{1761} - \frac{705817}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{- \frac{839119 y}{1761} - \frac{705817}{1761} + 697 - 625 y}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.5

To write $- 625 y$ as a fraction with a common denominator, multiply by $\frac{1761}{1761}$ .

$- \frac{- \frac{839119 y}{1761} - 625 y \cdot \frac{1761}{1761} - \frac{705817}{1761} + 697}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.6

Combine $- 625 y$ and $\frac{1761}{1761}$ .

$- \frac{- \frac{839119 y}{1761} + \frac{- 625 y \cdot 1761}{1761} - \frac{705817}{1761} + 697}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.7

Combine the numerators over the common denominator.

$- \frac{\frac{- 839119 y - 625 y \cdot 1761}{1761} - \frac{705817}{1761} + 697}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.8

Combine the numerators over the common denominator.

$- \frac{\frac{- 839119 y - 625 y \cdot 1761 - 705817}{1761} + 697}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.9

To write $697$ as a fraction with a common denominator, multiply by $\frac{1761}{1761}$ .

$- \frac{\frac{- 839119 y - 625 y \cdot 1761 - 705817}{1761} + 697 \cdot \frac{1761}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.10

Combine $697$ and $\frac{1761}{1761}$ .

$- \frac{\frac{- 839119 y - 625 y \cdot 1761 - 705817}{1761} + \frac{697 \cdot 1761}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.11

Combine the numerators over the common denominator.

$- \frac{\frac{- 839119 y - 625 y \cdot 1761 - 705817 + 697 \cdot 1761}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.12

Reorder terms.

$- \frac{\frac{- 625 \cdot (1761 y) - 839119 y + 697 \cdot 1761 - 705817}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13

Rewrite $\frac{- 625 \cdot 1761 y - 839119 y + 697 \cdot 1761 - 705817}{1761}$ in a factored form.

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Step 4.2.1.1.13.1

Multiply $- 625$ by $1761$ .

$- \frac{\frac{- 1100625 y - 839119 y + 697 \cdot 1761 - 705817}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.2

Multiply $697$ by $1761$ .

$- \frac{\frac{- 1100625 y - 839119 y + 1227417 - 705817}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.3

Subtract $839119 y$ from $- 1100625 y$ .

$- \frac{\frac{- 1939744 y + 1227417 - 705817}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.4

Subtract $705817$ from $1227417$ .

$- \frac{\frac{- 1939744 y + 521600}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.5

Factor $32$ out of $- 1939744 y + 521600$ .

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Step 4.2.1.1.13.5.1

Factor $32$ out of $- 1939744 y$ .

$- \frac{\frac{32 (- 60617 y) + 521600}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.5.2

Factor $32$ out of $521600$ .

$- \frac{\frac{32 (- 60617 y) + 32 (16300)}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.1.13.5.3

Factor $32$ out of $32 (- 60617 y) + 32 (16300)$ .

$- \frac{\frac{32 (- 60617 y + 16300)}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{\frac{32 (- 60617 y + 16300)}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{\frac{32 (- 60617 y + 16300)}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{\frac{32 (- 60617 y + 16300)}{1761}}{32} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{32 (- 60617 y + 16300)}{1761} \cdot \frac{1}{32}) = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.3

Cancel the common factor of $32$ .

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Step 4.2.1.3.1

Cancel the common factor.

$- (\frac{32 (- 60617 y + 16300)}{1761} \cdot \frac{1}{32}) = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.3.2

Rewrite the expression.

$- \frac{- 60617 y + 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$- \frac{- 60617 y + 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.4

Factor $- 1$ out of $- 60617 y$ .

$- \frac{- (60617 y) + 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.5

Rewrite $16300$ as $- 1 (- 16300)$ .

$- \frac{- (60617 y) - 1 \cdot - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.6

Factor $- 1$ out of $- (60617 y) - 1 (- 16300)$ .

$- \frac{- (60617 y - 16300)}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.7

Simplify the expression.

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Step 4.2.1.7.1

Rewrite $- (60617 y - 16300)$ as $- 1 (60617 y - 16300)$ .

$- \frac{- 1 (60617 y - 16300)}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.7.2

Move the negative in front of the fraction.

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.7.3

Multiply $- 1$ by $- 1$ .

$1 (\frac{60617 y - 16300}{1761}) = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.2.1.7.4

Multiply $\frac{60617 y - 16300}{1761}$ by $1$ .

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$

Step 4.3

Replace all occurrences of $z$ in $x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 z}{32}$ with $- \frac{491 y}{1761} - \frac{413}{1761}$ .

$x = - \frac{17}{32} + \frac{9 y}{32} - \frac{37 (- \frac{491 y}{1761} - \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- \frac{17}{32} + \frac{9 y}{32} - \frac{37 (- \frac{491 y}{1761} - \frac{413}{1761})}{32}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 17 + 9 y - 37 (- \frac{491 y}{1761} - \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{- 17 + 9 y - 37 (- \frac{491 y}{1761}) - 37 (- \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.2

Multiply $- 37 (- \frac{491 y}{1761})$ .

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Step 4.4.1.2.2.1

Multiply $- 1$ by $- 37$ .

$x = \frac{- 17 + 9 y + 37 (\frac{491 y}{1761}) - 37 (- \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.2.2

Combine $37$ and $\frac{491 y}{1761}$ .

$x = \frac{- 17 + 9 y + \frac{37 (491 y)}{1761} - 37 (- \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.2.3

Multiply $491$ by $37$ .

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} - 37 (- \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} - 37 (- \frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.3

Multiply $- 37 (- \frac{413}{1761})$ .

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Step 4.4.1.2.3.1

Multiply $- 1$ by $- 37$ .

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} + 37 (\frac{413}{1761})}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.3.2

Combine $37$ and $\frac{413}{1761}$ .

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} + \frac{37 \cdot 413}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.2.3.3

Multiply $37$ by $413$ .

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} + \frac{15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} + \frac{15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{- 17 + 9 y + \frac{18167 y}{1761} + \frac{15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.3

To write $- 17$ as a fraction with a common denominator, multiply by $\frac{1761}{1761}$ .

$x = \frac{9 y + \frac{18167 y}{1761} - 17 \cdot \frac{1761}{1761} + \frac{15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.4

Combine $- 17$ and $\frac{1761}{1761}$ .

$x = \frac{9 y + \frac{18167 y}{1761} + \frac{- 17 \cdot 1761}{1761} + \frac{15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{9 y + \frac{18167 y}{1761} + \frac{- 17 \cdot 1761 + 15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.6

Simplify the numerator.

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Step 4.4.1.6.1

Multiply $- 17$ by $1761$ .

$x = \frac{9 y + \frac{18167 y}{1761} + \frac{- 29937 + 15281}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.6.2

Add $- 29937$ and $15281$ .

$x = \frac{9 y + \frac{18167 y}{1761} + \frac{- 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{9 y + \frac{18167 y}{1761} + \frac{- 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.7

Move the negative in front of the fraction.

$x = \frac{9 y + \frac{18167 y}{1761} - \frac{14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.8

To write $9 y$ as a fraction with a common denominator, multiply by $\frac{1761}{1761}$ .

$x = \frac{9 y \cdot \frac{1761}{1761} + \frac{18167 y}{1761} - \frac{14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.9

Combine $9 y$ and $\frac{1761}{1761}$ .

$x = \frac{\frac{9 y \cdot 1761}{1761} + \frac{18167 y}{1761} - \frac{14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.10

Combine the numerators over the common denominator.

$x = \frac{\frac{9 y \cdot 1761 + 18167 y}{1761} - \frac{14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.11

Combine the numerators over the common denominator.

$x = \frac{\frac{9 y \cdot 1761 + 18167 y - 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.12

Multiply $1761$ by $9$ .

$x = \frac{\frac{15849 y + 18167 y - 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.13

Add $15849 y$ and $18167 y$ .

$x = \frac{\frac{34016 y - 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.14

Factor $32$ out of $34016 y - 14656$ .

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Step 4.4.1.14.1

Factor $32$ out of $34016 y$ .

$x = \frac{\frac{32 (1063 y) - 14656}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.14.2

Factor $32$ out of $- 14656$ .

$x = \frac{\frac{32 (1063 y) + 32 (- 458)}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.14.3

Factor $32$ out of $32 (1063 y) + 32 (- 458)$ .

$x = \frac{\frac{32 (1063 y - 458)}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{\frac{32 (1063 y - 458)}{1761}}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.15

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{32 (1063 y - 458)}{1761} \cdot \frac{1}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.16

Cancel the common factor of $32$ .

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Step 4.4.1.16.1

Cancel the common factor.

$x = \frac{32 (1063 y - 458)}{1761} \cdot \frac{1}{32}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 4.4.1.16.2

Rewrite the expression.

$x = \frac{1063 y - 458}{1761}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{1063 y - 458}{1761}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{1063 y - 458}{1761}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{1063 y - 458}{1761}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{1063 y - 458}{1761}$

$\frac{60617 y - 16300}{1761} = 40$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5

Solve for $y$ in $\frac{60617 y - 16300}{1761} = 40$ .

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Step 5.1

Multiply both sides by $1761$ .

$\frac{60617 y - 16300}{1761} \cdot 1761 = 40 \cdot 1761$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

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 $1761$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{60617 y - 16300}{1761} \cdot 1761 = 40 \cdot 1761$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.2.1.1.2

Rewrite the expression.

$60617 y - 16300 = 40 \cdot 1761$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$60617 y - 16300 = 40 \cdot 1761$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$60617 y - 16300 = 40 \cdot 1761$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $40$ by $1761$ .

$60617 y - 16300 = 70440$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$60617 y - 16300 = 70440$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$60617 y - 16300 = 70440$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.3

Solve for $y$ .

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Step 5.3.1

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

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Step 5.3.1.1

Add $16300$ to both sides of the equation.

$60617 y = 70440 + 16300$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.3.1.2

Add $70440$ and $16300$ .

$60617 y = 86740$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$60617 y = 86740$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.3.2

Divide each term in $60617 y = 86740$ by $60617$ and simplify.

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Step 5.3.2.1

Divide each term in $60617 y = 86740$ by $60617$ .

$\frac{60617 y}{60617} = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $60617$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{60617 y}{60617} = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 5.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$y = \frac{86740}{60617}$

$x = \frac{1063 y - 458}{1761}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6

Replace all occurrences of $y$ with $\frac{86740}{60617}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{1063 y - 458}{1761}$ with $\frac{86740}{60617}$ .

$x = \frac{1063 (\frac{86740}{60617}) - 458}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{1063 (\frac{86740}{60617}) - 458}{1761}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $1063 (\frac{86740}{60617})$ .

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Step 6.2.1.1.1.1

Combine $1063$ and $\frac{86740}{60617}$ .

$x = \frac{\frac{1063 \cdot 86740}{60617} - 458}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.1.2

Multiply $1063$ by $86740$ .

$x = \frac{\frac{92204620}{60617} - 458}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{\frac{92204620}{60617} - 458}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.2

To write $- 458$ as a fraction with a common denominator, multiply by $\frac{60617}{60617}$ .

$x = \frac{\frac{92204620}{60617} - 458 \cdot \frac{60617}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.3

Combine $- 458$ and $\frac{60617}{60617}$ .

$x = \frac{\frac{92204620}{60617} + \frac{- 458 \cdot 60617}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.4

Combine the numerators over the common denominator.

$x = \frac{\frac{92204620 - 458 \cdot 60617}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.5

Simplify the numerator.

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Step 6.2.1.1.5.1

Multiply $- 458$ by $60617$ .

$x = \frac{\frac{92204620 - 27762586}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.1.5.2

Subtract $27762586$ from $92204620$ .

$x = \frac{\frac{64442034}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{\frac{64442034}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{\frac{64442034}{60617}}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{64442034}{60617} \cdot \frac{1}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.3

Cancel the common factor of $1761$ .

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Step 6.2.1.3.1

Factor $1761$ out of $64442034$ .

$x = \frac{1761 (36594)}{60617} \cdot \frac{1}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.3.2

Cancel the common factor.

$x = \frac{1761 \cdot 36594}{60617} \cdot \frac{1}{1761}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.2.1.3.3

Rewrite the expression.

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{491 y}{1761} - \frac{413}{1761}$

Step 6.3

Replace all occurrences of $y$ in $z = - \frac{491 y}{1761} - \frac{413}{1761}$ with $\frac{86740}{60617}$ .

$z = - \frac{491 (\frac{86740}{60617})}{1761} - \frac{413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{491 (\frac{86740}{60617})}{1761} - \frac{413}{1761}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$z = \frac{- 491 (\frac{86740}{60617}) - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.2

Simplify each term.

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Step 6.4.1.2.1

Multiply $- 491 (\frac{86740}{60617})$ .

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Step 6.4.1.2.1.1

Combine $- 491$ and $\frac{86740}{60617}$ .

$z = \frac{\frac{- 491 \cdot 86740}{60617} - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.2.1.2

Multiply $- 491$ by $86740$ .

$z = \frac{\frac{- 42589340}{60617} - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = \frac{\frac{- 42589340}{60617} - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.2.2

Move the negative in front of the fraction.

$z = \frac{- \frac{42589340}{60617} - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = \frac{- \frac{42589340}{60617} - 413}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.3

To write $- 413$ as a fraction with a common denominator, multiply by $\frac{60617}{60617}$ .

$z = \frac{- \frac{42589340}{60617} - 413 \cdot \frac{60617}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.4

Combine $- 413$ and $\frac{60617}{60617}$ .

$z = \frac{- \frac{42589340}{60617} + \frac{- 413 \cdot 60617}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.5

Combine the numerators over the common denominator.

$z = \frac{\frac{- 42589340 - 413 \cdot 60617}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.6

Simplify the numerator.

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Step 6.4.1.6.1

Multiply $- 413$ by $60617$ .

$z = \frac{\frac{- 42589340 - 25034821}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.6.2

Subtract $25034821$ from $- 42589340$ .

$z = \frac{\frac{- 67624161}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = \frac{\frac{- 67624161}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.7

Move the negative in front of the fraction.

$z = \frac{- \frac{67624161}{60617}}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.8

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{67624161}{60617} \cdot \frac{1}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.9

Cancel the common factor of $1761$ .

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Step 6.4.1.9.1

Move the leading negative in $- \frac{67624161}{60617}$ into the numerator.

$z = \frac{- 67624161}{60617} \cdot \frac{1}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.9.2

Factor $1761$ out of $- 67624161$ .

$z = \frac{1761 (- 38401)}{60617} \cdot \frac{1}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.9.3

Cancel the common factor.

$z = \frac{1761 \cdot - 38401}{60617} \cdot \frac{1}{1761}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.9.4

Rewrite the expression.

$z = \frac{- 38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = \frac{- 38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 6.4.1.10

Move the negative in front of the fraction.

$z = - \frac{38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

$z = - \frac{38401}{60617}$

$x = \frac{36594}{60617}$

$y = \frac{86740}{60617}$

Step 7

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

$(\frac{36594}{60617}, \frac{86740}{60617}, - \frac{38401}{60617})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{36594}{60617}, \frac{86740}{60617}, - \frac{38401}{60617})$

Equation Form:

$x = \frac{36594}{60617}, y = \frac{86740}{60617}, z = - \frac{38401}{60617}$