Finite Math Examples

$x + y + z = 57$ , $16 x + 13 y + 15 x = 847$ , $18 x + 14 y + 15 z = 900$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 57 - y$

$31 x + 13 y = 847$

$18 x + 14 y + 15 z = 900$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 57 - y - z$

$31 x + 13 y = 847$

$18 x + 14 y + 15 z = 900$

$x = 57 - y - z$

$31 x + 13 y = 847$

$18 x + 14 y + 15 z = 900$

Step 2

Replace all occurrences of $x$ with $57 - y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $31 x + 13 y = 847$ with $57 - y - z$ .

$31 (57 - y - z) + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $31 (57 - y - z) + 13 y$ .

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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.

$31 \cdot 57 + 31 (- y) + 31 (- z) + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $31$ by $57$ .

$1767 + 31 (- y) + 31 (- z) + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.2.1.1.2.2

Multiply $- 1$ by $31$ .

$1767 - 31 y + 31 (- z) + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.2.1.1.2.3

Multiply $- 1$ by $31$ .

$1767 - 31 y - 31 z + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

$1767 - 31 y - 31 z + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

$1767 - 31 y - 31 z + 13 y = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.2.1.2

Add $- 31 y$ and $13 y$ .

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$18 x + 14 y + 15 z = 900$

Step 2.3

Replace all occurrences of $x$ in $18 x + 14 y + 15 z = 900$ with $57 - y - z$ .

$18 (57 - y - z) + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $18 (57 - y - z) + 14 y + 15 z$ .

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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.

$18 \cdot 57 + 18 (- y) + 18 (- z) + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $18$ by $57$ .

$1026 + 18 (- y) + 18 (- z) + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $18$ .

$1026 - 18 y + 18 (- z) + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $18$ .

$1026 - 18 y - 18 z + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 18 y - 18 z + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 18 y - 18 z + 14 y + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 18 y$ and $14 y$ .

$1026 - 4 y - 18 z + 15 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 2.4.1.2.2

Add $- 18 z$ and $15 z$ .

$1026 - 4 y - 3 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 4 y - 3 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 4 y - 3 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 4 y - 3 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$1026 - 4 y - 3 z = 900$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3

Solve for $y$ in $1026 - 4 y - 3 z = 900$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1026$ from both sides of the equation.

$- 4 y - 3 z = 900 - 1026$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.1.2

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

$- 4 y = 900 - 1026 + 3 z$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.1.3

Subtract $1026$ from $900$ .

$- 4 y = - 126 + 3 z$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$- 4 y = - 126 + 3 z$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2

Divide each term in $- 4 y = - 126 + 3 z$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 y = - 126 + 3 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 126}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 126}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 126}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{- 126}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{- 126}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

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 $- 126$ and $- 4$ .

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Step 3.2.3.1.1.1

Factor $- 2$ out of $- 126$ .

$y = \frac{- 2 \cdot 63}{- 4} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $- 2$ out of $- 4$ .

$y = \frac{- 2 \cdot 63}{- 2 \cdot 2} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 63}{- 2 \cdot 2} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{63}{2} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} + \frac{3 z}{- 4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = \frac{63}{2} - \frac{3 z}{4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} - \frac{3 z}{4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} - \frac{3 z}{4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} - \frac{3 z}{4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

$y = \frac{63}{2} - \frac{3 z}{4}$

$1767 - 18 y - 31 z = 847$

$x = 57 - y - z$

Step 4

Replace all occurrences of $y$ with $\frac{63}{2} - \frac{3 z}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $1767 - 18 y - 31 z = 847$ with $\frac{63}{2} - \frac{3 z}{4}$ .

$1767 - 18 (\frac{63}{2} - \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $1767 - 18 (\frac{63}{2} - \frac{3 z}{4}) - 31 z$ .

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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.

$1767 - 18 (\frac{63}{2}) - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Factor $2$ out of $- 18$ .

$1767 + 2 (- 9) (\frac{63}{2}) - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.2.2

Cancel the common factor.

$1767 + 2 \cdot (- 9 (\frac{63}{2})) - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.2.3

Rewrite the expression.

$1767 - 9 \cdot 63 - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1767 - 9 \cdot 63 - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.3

Multiply $- 9$ by $63$ .

$1767 - 567 - 18 (- \frac{3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.4.1

Move the leading negative in $- \frac{3 z}{4}$ into the numerator.

$1767 - 567 - 18 \frac{- 3 z}{4} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.4.2

Factor $2$ out of $- 18$ .

$1767 - 567 + 2 (- 9) (\frac{- 3 z}{4}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.4.3

Factor $2$ out of $4$ .

$1767 - 567 + 2 \cdot (- 9 \frac{- 3 z}{2 \cdot 2}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.4.4

Cancel the common factor.

$1767 - 567 + 2 \cdot (- 9 \frac{- 3 z}{2 \cdot 2}) - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.4.5

Rewrite the expression.

$1767 - 567 - 9 \frac{- 3 z}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1767 - 567 - 9 \frac{- 3 z}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.5

Combine $- 9$ and $\frac{- 3 z}{2}$ .

$1767 - 567 + \frac{- 9 (- 3 z)}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.1.6

Multiply $- 3$ by $- 9$ .

$1767 - 567 + \frac{27 z}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1767 - 567 + \frac{27 z}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.2

Subtract $567$ from $1767$ .

$1200 + \frac{27 z}{2} - 31 z = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.3

To write $- 31 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1200 + \frac{27 z}{2} - 31 z \cdot \frac{2}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.4

Simplify terms.

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Step 4.2.1.4.1

Combine $- 31 z$ and $\frac{2}{2}$ .

$1200 + \frac{27 z}{2} + \frac{- 31 z \cdot 2}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.4.2

Combine the numerators over the common denominator.

$1200 + \frac{27 z - 31 z \cdot 2}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 + \frac{27 z - 31 z \cdot 2}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5

Simplify each term.

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Step 4.2.1.5.1

Simplify the numerator.

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Step 4.2.1.5.1.1

Factor $z$ out of $27 z - 31 z \cdot 2$ .

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Step 4.2.1.5.1.1.1

Factor $z$ out of $27 z$ .

$1200 + \frac{z \cdot 27 - 31 z \cdot 2}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.1.1.2

Factor $z$ out of $- 31 z \cdot 2$ .

$1200 + \frac{z \cdot 27 + z (- 31 \cdot 2)}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.1.1.3

Factor $z$ out of $z \cdot 27 + z (- 31 \cdot 2)$ .

$1200 + \frac{z (27 - 31 \cdot 2)}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 + \frac{z (27 - 31 \cdot 2)}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.1.2

Multiply $- 31$ by $2$ .

$1200 + \frac{z (27 - 62)}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.1.3

Subtract $62$ from $27$ .

$1200 + \frac{z \cdot - 35}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 + \frac{z \cdot - 35}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.2

Move $- 35$ to the left of $z$ .

$1200 + \frac{- 35 \cdot z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.2.1.5.3

Move the negative in front of the fraction.

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 57 - y - z$ with $\frac{63}{2} - \frac{3 z}{4}$ .

$x = 57 - (\frac{63}{2} - \frac{3 z}{4}) - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $57 - (\frac{63}{2} - \frac{3 z}{4}) - z$ .

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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.

$x = 57 - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.1.2

Multiply $- - \frac{3 z}{4}$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$x = 57 - \frac{63}{2} + 1 (\frac{3 z}{4}) - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.1.2.2

Multiply $\frac{3 z}{4}$ by $1$ .

$x = 57 - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = 57 - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.2

To write $57$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 57 \cdot \frac{2}{2} - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.3

Combine $57$ and $\frac{2}{2}$ .

$x = \frac{57 \cdot 2}{2} - \frac{63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{57 \cdot 2 - 63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $57$ by $2$ .

$x = \frac{114 - 63}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.5.2

Subtract $63$ from $114$ .

$x = \frac{51}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} + \frac{3 z}{4} - z$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.6

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

$x = \frac{51}{2} + \frac{3 z}{4} - z \cdot \frac{4}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.7

Simplify terms.

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Step 4.4.1.7.1

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

$x = \frac{51}{2} + \frac{3 z}{4} + \frac{- z \cdot 4}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.7.2

Combine the numerators over the common denominator.

$x = \frac{51}{2} + \frac{3 z - z \cdot 4}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} + \frac{3 z - z \cdot 4}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

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 $3 z - z \cdot 4$ .

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Step 4.4.1.8.1.1.1

Factor $z$ out of $3 z$ .

$x = \frac{51}{2} + \frac{z \cdot 3 - z \cdot 4}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.1.1.2

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

$x = \frac{51}{2} + \frac{z \cdot 3 + z (- 1 \cdot 4)}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.1.1.3

Factor $z$ out of $z \cdot 3 + z (- 1 \cdot 4)$ .

$x = \frac{51}{2} + \frac{z (3 - 1 \cdot 4)}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} + \frac{z (3 - 1 \cdot 4)}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.1.2

Multiply $- 1$ by $4$ .

$x = \frac{51}{2} + \frac{z (3 - 4)}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.1.3

Subtract $4$ from $3$ .

$x = \frac{51}{2} + \frac{z \cdot - 1}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} + \frac{z \cdot - 1}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.2

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

$x = \frac{51}{2} + \frac{- 1 \cdot z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 4.4.1.8.3

Move the negative in front of the fraction.

$x = \frac{51}{2} - \frac{z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - \frac{z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - \frac{z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - \frac{z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - \frac{z}{4}$

$1200 - \frac{35 z}{2} = 847$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5

Solve for $z$ in $1200 - \frac{35 z}{2} = 847$ .

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Step 5.1

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

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Step 5.1.1

Subtract $1200$ from both sides of the equation.

$- \frac{35 z}{2} = 847 - 1200$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.1.2

Subtract $1200$ from $847$ .

$- \frac{35 z}{2} = - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$- \frac{35 z}{2} = - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.2

Multiply both sides of the equation by $- \frac{2}{35}$ .

$- \frac{2}{35} \cdot (- \frac{35 z}{2}) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $- \frac{2}{35} (- \frac{35 z}{2})$ .

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Step 5.3.1.1.1

Cancel the common factor of $2$ .

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Step 5.3.1.1.1.1

Move the leading negative in $- \frac{2}{35}$ into the numerator.

$\frac{- 2}{35} \cdot (- \frac{35 z}{2}) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.1.2

Move the leading negative in $- \frac{35 z}{2}$ into the numerator.

$\frac{- 2}{35} \cdot \frac{- 35 z}{2} = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{35} \cdot \frac{- 35 z}{2} = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{35} \cdot \frac{- 35 z}{2} = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{35} \cdot (- 35 z) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$\frac{- 1}{35} \cdot (- 35 z) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.2

Cancel the common factor of $35$ .

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Step 5.3.1.1.2.1

Factor $35$ out of $- 35 z$ .

$\frac{- 1}{35} \cdot (35 (- z)) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{35} \cdot (35 (- z)) = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.2.3

Rewrite the expression.

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.3

Multiply.

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Step 5.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.1.1.3.2

Multiply $z$ by $1$ .

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = - \frac{2}{35} \cdot - 353$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $- \frac{2}{35} \cdot - 353$ .

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Step 5.3.2.1.1

Multiply $- 353$ by $- 1$ .

$z = 353 (\frac{2}{35})$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.2.1.2

Combine $353$ and $\frac{2}{35}$ .

$z = \frac{353 \cdot 2}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 5.3.2.1.3

Multiply $353$ by $2$ .

$z = \frac{706}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = \frac{706}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = \frac{706}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = \frac{706}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$z = \frac{706}{35}$

$x = \frac{51}{2} - \frac{z}{4}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6

Replace all occurrences of $z$ with $\frac{706}{35}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{51}{2} - \frac{z}{4}$ with $\frac{706}{35}$ .

$x = \frac{51}{2} - \frac{\frac{706}{35}}{4}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{51}{2} - \frac{\frac{706}{35}}{4}$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{51}{2} - (\frac{706}{35} \cdot \frac{1}{4})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.2.1

Factor $2$ out of $706$ .

$x = \frac{51}{2} - (\frac{2 (353)}{35} \cdot \frac{1}{4})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.2.2

Factor $2$ out of $4$ .

$x = \frac{51}{2} - (\frac{2 \cdot 353}{35} \cdot \frac{1}{2 \cdot 2})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.2.3

Cancel the common factor.

$x = \frac{51}{2} - (\frac{2 \cdot 353}{35} \cdot \frac{1}{2 \cdot 2})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.2.4

Rewrite the expression.

$x = \frac{51}{2} - (\frac{353}{35} \cdot \frac{1}{2})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - (\frac{353}{35} \cdot \frac{1}{2})$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.3

Multiply $\frac{353}{35}$ by $\frac{1}{2}$ .

$x = \frac{51}{2} - \frac{353}{35 \cdot 2}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.1.4

Multiply $35$ by $2$ .

$x = \frac{51}{2} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51}{2} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.2

To write $\frac{51}{2}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$x = \frac{51}{2} \cdot \frac{35}{35} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.3

Write each expression with a common denominator of $70$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.1.3.1

Multiply $\frac{51}{2}$ by $\frac{35}{35}$ .

$x = \frac{51 \cdot 35}{2 \cdot 35} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.3.2

Multiply $2$ by $35$ .

$x = \frac{51 \cdot 35}{70} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{51 \cdot 35}{70} - \frac{353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.4

Combine the numerators over the common denominator.

$x = \frac{51 \cdot 35 - 353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $51$ by $35$ .

$x = \frac{1785 - 353}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.5.2

Subtract $353$ from $1785$ .

$x = \frac{1432}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{1432}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.6

Cancel the common factor of $1432$ and $70$ .

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Step 6.2.1.6.1

Factor $2$ out of $1432$ .

$x = \frac{2 (716)}{70}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.6.2

Cancel the common factors.

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Step 6.2.1.6.2.1

Factor $2$ out of $70$ .

$x = \frac{2 \cdot 716}{2 \cdot 35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot 716}{2 \cdot 35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.2.1.6.2.3

Rewrite the expression.

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{3 z}{4}$

Step 6.3

Replace all occurrences of $z$ in $y = \frac{63}{2} - \frac{3 z}{4}$ with $\frac{706}{35}$ .

$y = \frac{63}{2} - \frac{3 (\frac{706}{35})}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{63}{2} - \frac{3 (\frac{706}{35})}{4}$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Combine $3$ and $\frac{706}{35}$ .

$y = \frac{63}{2} - \frac{\frac{3 \cdot 706}{35}}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.2

Multiply $3$ by $706$ .

$y = \frac{63}{2} - \frac{\frac{2118}{35}}{4}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{63}{2} - (\frac{2118}{35} \cdot \frac{1}{4})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.4

Cancel the common factor of $2$ .

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Step 6.4.1.1.4.1

Factor $2$ out of $2118$ .

$y = \frac{63}{2} - (\frac{2 (1059)}{35} \cdot \frac{1}{4})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.4.2

Factor $2$ out of $4$ .

$y = \frac{63}{2} - (\frac{2 \cdot 1059}{35} \cdot \frac{1}{2 \cdot 2})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.4.3

Cancel the common factor.

$y = \frac{63}{2} - (\frac{2 \cdot 1059}{35} \cdot \frac{1}{2 \cdot 2})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.4.4

Rewrite the expression.

$y = \frac{63}{2} - (\frac{1059}{35} \cdot \frac{1}{2})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - (\frac{1059}{35} \cdot \frac{1}{2})$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.5

Multiply $\frac{1059}{35}$ by $\frac{1}{2}$ .

$y = \frac{63}{2} - \frac{1059}{35 \cdot 2}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.1.6

Multiply $35$ by $2$ .

$y = \frac{63}{2} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63}{2} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.2

To write $\frac{63}{2}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$y = \frac{63}{2} \cdot \frac{35}{35} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.3

Write each expression with a common denominator of $70$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.1.3.1

Multiply $\frac{63}{2}$ by $\frac{35}{35}$ .

$y = \frac{63 \cdot 35}{2 \cdot 35} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.3.2

Multiply $2$ by $35$ .

$y = \frac{63 \cdot 35}{70} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{63 \cdot 35}{70} - \frac{1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$y = \frac{63 \cdot 35 - 1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $63$ by $35$ .

$y = \frac{2205 - 1059}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.5.2

Subtract $1059$ from $2205$ .

$y = \frac{1146}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{1146}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.6

Cancel the common factor of $1146$ and $70$ .

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Step 6.4.1.6.1

Factor $2$ out of $1146$ .

$y = \frac{2 (573)}{70}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.6.2

Cancel the common factors.

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Step 6.4.1.6.2.1

Factor $2$ out of $70$ .

$y = \frac{2 \cdot 573}{2 \cdot 35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.6.2.2

Cancel the common factor.

$y = \frac{2 \cdot 573}{2 \cdot 35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 6.4.1.6.2.3

Rewrite the expression.

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

$y = \frac{573}{35}$

$x = \frac{716}{35}$

$z = \frac{706}{35}$

Step 7

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

$(\frac{716}{35}, \frac{573}{35}, \frac{706}{35})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{716}{35}, \frac{573}{35}, \frac{706}{35})$

Equation Form:

$x = \frac{716}{35}, y = \frac{573}{35}, z = \frac{706}{35}$