Finite Math Examples

$x - y + z = 4$ , $2 x - 8 y - 3 z = 24$ , $3 x + 4 y + 2 x = 7$

Step 1

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

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Step 1.1

Add $y$ to both sides of the equation.

$x + z = 4 + y$

$2 x - 8 y - 3 z = 24$

$5 x + 4 y = 7$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 4 + y - z$

$2 x - 8 y - 3 z = 24$

$5 x + 4 y = 7$

$x = 4 + y - z$

$2 x - 8 y - 3 z = 24$

$5 x + 4 y = 7$

Step 2

Replace all occurrences of $x$ with $4 + y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 8 y - 3 z = 24$ with $4 + y - z$ .

$2 (4 + y - z) - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (4 + y - z) - 8 y - 3 z$ .

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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.

$2 \cdot 4 + 2 y + 2 (- z) - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $2$ by $4$ .

$8 + 2 y + 2 (- z) - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.2.1.1.2.2

Multiply $- 1$ by $2$ .

$8 + 2 y - 2 z - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

$8 + 2 y - 2 z - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

$8 + 2 y - 2 z - 8 y - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $8 y$ from $2 y$ .

$8 - 6 y - 2 z - 3 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.2.1.2.2

Subtract $3 z$ from $- 2 z$ .

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$5 x + 4 y = 7$

Step 2.3

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

$5 (4 + y - z) + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (4 + y - z) + 4 y$ .

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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.

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

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $5$ by $4$ .

$20 + 5 y + 5 (- z) + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $5$ .

$20 + 5 y - 5 z + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$20 + 5 y - 5 z + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$20 + 5 y - 5 z + 4 y = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 2.4.1.2

Add $5 y$ and $4 y$ .

$20 + 9 y - 5 z = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$20 + 9 y - 5 z = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$20 + 9 y - 5 z = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$20 + 9 y - 5 z = 7$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3

Solve for $y$ in $20 + 9 y - 5 z = 7$ .

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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 $20$ from both sides of the equation.

$9 y - 5 z = 7 - 20$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.1.2

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

$9 y = 7 - 20 + 5 z$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.1.3

Subtract $20$ from $7$ .

$9 y = - 13 + 5 z$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$9 y = - 13 + 5 z$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.2

Divide each term in $9 y = - 13 + 5 z$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 y = - 13 + 5 z$ by $9$ .

$\frac{9 y}{9} = \frac{- 13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{- 13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$y = \frac{- 13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$y = \frac{- 13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$8 - 6 y - 5 z = 24$

$x = 4 + y - z$

Step 4

Replace all occurrences of $y$ with $- \frac{13}{9} + \frac{5 z}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $8 - 6 y - 5 z = 24$ with $- \frac{13}{9} + \frac{5 z}{9}$ .

$8 - 6 (- \frac{13}{9} + \frac{5 z}{9}) - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $8 - 6 (- \frac{13}{9} + \frac{5 z}{9}) - 5 z$ .

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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.

$8 - 6 (- \frac{13}{9}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{13}{9}$ into the numerator.

$8 - 6 (\frac{- 13}{9}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.2.2

Factor $3$ out of $- 6$ .

$8 + 3 (- 2) (\frac{- 13}{9}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.2.3

Factor $3$ out of $9$ .

$8 + 3 \cdot (- 2 \frac{- 13}{3 \cdot 3}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.2.4

Cancel the common factor.

$8 + 3 \cdot (- 2 \frac{- 13}{3 \cdot 3}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.2.5

Rewrite the expression.

$8 - 2 (\frac{- 13}{3}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$8 - 2 (\frac{- 13}{3}) - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.3

Combine $- 2$ and $\frac{- 13}{3}$ .

$8 + \frac{- 2 \cdot - 13}{3} - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.4

Multiply $- 2$ by $- 13$ .

$8 + \frac{26}{3} - 6 \frac{5 z}{9} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.5

Cancel the common factor of $3$ .

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Step 4.2.1.1.5.1

Factor $3$ out of $- 6$ .

$8 + \frac{26}{3} + 3 (- 2) (\frac{5 z}{9}) - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.5.2

Factor $3$ out of $9$ .

$8 + \frac{26}{3} + 3 \cdot (- 2 \frac{5 z}{3 \cdot 3}) - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.5.3

Cancel the common factor.

$8 + \frac{26}{3} + 3 \cdot (- 2 \frac{5 z}{3 \cdot 3}) - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.5.4

Rewrite the expression.

$8 + \frac{26}{3} - 2 \frac{5 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$8 + \frac{26}{3} - 2 \frac{5 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.6

Combine $- 2$ and $\frac{5 z}{3}$ .

$8 + \frac{26}{3} + \frac{- 2 (5 z)}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.7

Multiply $5$ by $- 2$ .

$8 + \frac{26}{3} + \frac{- 10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.1.8

Move the negative in front of the fraction.

$8 + \frac{26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$8 + \frac{26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} + \frac{26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.3

Combine $8$ and $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3} + \frac{26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 + 26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $8$ by $3$ .

$\frac{24 + 26}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.5.2

Add $24$ and $26$ .

$\frac{50}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$\frac{50}{3} - \frac{10 z}{3} - 5 z = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.6

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

$\frac{50}{3} - \frac{10 z}{3} - 5 z \cdot \frac{3}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.7

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

$\frac{50}{3} - \frac{10 z}{3} + \frac{- 5 z \cdot 3}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.8

Combine the numerators over the common denominator.

$\frac{50}{3} + \frac{- 10 z - 5 z \cdot 3}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{50 - 10 z - 5 z \cdot 3}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.10

Multiply $3$ by $- 5$ .

$\frac{50 - 10 z - 15 z}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.11

Subtract $15 z$ from $- 10 z$ .

$\frac{50 - 25 z}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.12

Factor $25$ out of $50 - 25 z$ .

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Step 4.2.1.12.1

Factor $25$ out of $50$ .

$\frac{25 (2) - 25 z}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.12.2

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

$\frac{25 (2) + 25 (- z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.2.1.12.3

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

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 + y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 4 + y - z$ with $- \frac{13}{9} + \frac{5 z}{9}$ .

$x = 4 - \frac{13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4

Simplify $x = 4 - \frac{13}{9} + \frac{5 z}{9} - z$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$x = 4 - \frac{13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = 4 - \frac{13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $4 - \frac{13}{9} + \frac{5 z}{9} - z$ .

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Step 4.4.2.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = 4 \cdot \frac{9}{9} - \frac{13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.2

Combine $4$ and $\frac{9}{9}$ .

$x = \frac{4 \cdot 9}{9} - \frac{13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.3

Combine the numerators over the common denominator.

$x = \frac{4 \cdot 9 - 13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.4

Simplify the numerator.

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Step 4.4.2.1.4.1

Multiply $4$ by $9$ .

$x = \frac{36 - 13}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.4.2

Subtract $13$ from $36$ .

$x = \frac{23}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23}{9} + \frac{5 z}{9} - z$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.5

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

$x = \frac{23}{9} + \frac{5 z}{9} - z \cdot \frac{9}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.6

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

$x = \frac{23}{9} + \frac{5 z}{9} + \frac{- z \cdot 9}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.7

Combine the numerators over the common denominator.

$x = \frac{23}{9} + \frac{5 z - z \cdot 9}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.8

Combine the numerators over the common denominator.

$x = \frac{23 + 5 z - z \cdot 9}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.9

Multiply $9$ by $- 1$ .

$x = \frac{23 + 5 z - 9 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 4.4.2.1.10

Subtract $9 z$ from $5 z$ .

$x = \frac{23 - 4 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23 - 4 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23 - 4 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23 - 4 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23 - 4 z}{9}$

$\frac{25 (2 - z)}{3} = 24$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5

Solve for $z$ in $\frac{25 (2 - z)}{3} = 24$ .

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Step 5.1

Multiply both sides by $3$ .

$\frac{25 (2 - z)}{3} \cdot 3 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

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

Simplify $\frac{25 (2 - z)}{3} \cdot 3$ .

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Step 5.2.1.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{25 (2 - z)}{3} \cdot 3 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.1.1.1.2

Rewrite the expression.

$25 (2 - z) = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$25 (2 - z) = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.1.1.2

Apply the distributive property.

$25 \cdot 2 + 25 (- z) = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.1.1.3

Simplify the expression.

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Step 5.2.1.1.3.1

Multiply $25$ by $2$ .

$50 + 25 (- z) = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.1.1.3.2

Multiply $- 1$ by $25$ .

$50 - 25 z = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.1.1.3.3

Reorder $50$ and $- 25 z$ .

$- 25 z + 50 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z + 50 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z + 50 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z + 50 = 24 \cdot 3$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $24$ by $3$ .

$- 25 z + 50 = 72$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z + 50 = 72$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z + 50 = 72$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

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

Subtract $50$ from both sides of the equation.

$- 25 z = 72 - 50$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.3.1.2

Subtract $50$ from $72$ .

$- 25 z = 22$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$- 25 z = 22$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.3.2

Divide each term in $- 25 z = 22$ by $- 25$ and simplify.

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Step 5.3.2.1

Divide each term in $- 25 z = 22$ by $- 25$ .

$\frac{- 25 z}{- 25} = \frac{22}{- 25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $- 25$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{- 25 z}{- 25} = \frac{22}{- 25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{22}{- 25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = \frac{22}{- 25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = \frac{22}{- 25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Move the negative in front of the fraction.

$z = - \frac{22}{25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = - \frac{22}{25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = - \frac{22}{25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = - \frac{22}{25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$z = - \frac{22}{25}$

$x = \frac{23 - 4 z}{9}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6

Replace all occurrences of $z$ with $- \frac{22}{25}$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{23 - 4 z}{9}$ with $- \frac{22}{25}$ .

$x = \frac{23 - 4 (- \frac{22}{25})}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{23 - 4 (- \frac{22}{25})}{9}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 4 (- \frac{22}{25})$ .

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Step 6.2.1.1.1.1

Multiply $- 1$ by $- 4$ .

$x = \frac{23 + 4 (\frac{22}{25})}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.1.2

Combine $4$ and $\frac{22}{25}$ .

$x = \frac{23 + \frac{4 \cdot 22}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.1.3

Multiply $4$ by $22$ .

$x = \frac{23 + \frac{88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{23 + \frac{88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.2

To write $23$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$x = \frac{23 \cdot \frac{25}{25} + \frac{88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.3

Combine $23$ and $\frac{25}{25}$ .

$x = \frac{\frac{23 \cdot 25}{25} + \frac{88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.4

Combine the numerators over the common denominator.

$x = \frac{\frac{23 \cdot 25 + 88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.5

Simplify the numerator.

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Step 6.2.1.1.5.1

Multiply $23$ by $25$ .

$x = \frac{\frac{575 + 88}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.1.5.2

Add $575$ and $88$ .

$x = \frac{\frac{663}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{\frac{663}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{\frac{663}{25}}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{663}{25} \cdot \frac{1}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.3

Cancel the common factor of $3$ .

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Step 6.2.1.3.1

Factor $3$ out of $663$ .

$x = \frac{3 (221)}{25} \cdot \frac{1}{9}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.3.2

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 221}{25} \cdot \frac{1}{3 \cdot 3}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.3.3

Cancel the common factor.

$x = \frac{3 \cdot 221}{25} \cdot \frac{1}{3 \cdot 3}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.3.4

Rewrite the expression.

$x = \frac{221}{25} \cdot \frac{1}{3}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{221}{25} \cdot \frac{1}{3}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.4

Multiply $\frac{221}{25}$ by $\frac{1}{3}$ .

$x = \frac{221}{25 \cdot 3}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.2.1.5

Multiply $25$ by $3$ .

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{13}{9} + \frac{5 z}{9}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{13}{9} + \frac{5 z}{9}$ with $- \frac{22}{25}$ .

$y = - \frac{13}{9} + \frac{5 (- \frac{22}{25})}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{13}{9} + \frac{5 (- \frac{22}{25})}{9}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 13 + 5 (- \frac{22}{25})}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.2

Simplify each term.

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Step 6.4.1.2.1

Cancel the common factor of $5$ .

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Step 6.4.1.2.1.1

Move the leading negative in $- \frac{22}{25}$ into the numerator.

$y = \frac{- 13 + 5 (\frac{- 22}{25})}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.2.1.2

Factor $5$ out of $25$ .

$y = \frac{- 13 + 5 (\frac{- 22}{5 (5)})}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.2.1.3

Cancel the common factor.

$y = \frac{- 13 + 5 (\frac{- 22}{5 \cdot 5})}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.2.1.4

Rewrite the expression.

$y = \frac{- 13 + \frac{- 22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = \frac{- 13 + \frac{- 22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.2.2

Move the negative in front of the fraction.

$y = \frac{- 13 - \frac{22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = \frac{- 13 - \frac{22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.3

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{- 13 \cdot \frac{5}{5} - \frac{22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.4

Combine $- 13$ and $\frac{5}{5}$ .

$y = \frac{\frac{- 13 \cdot 5}{5} - \frac{22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.5

Combine the numerators over the common denominator.

$y = \frac{\frac{- 13 \cdot 5 - 22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.6

Simplify the numerator.

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Step 6.4.1.6.1

Multiply $- 13$ by $5$ .

$y = \frac{\frac{- 65 - 22}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.6.2

Subtract $22$ from $- 65$ .

$y = \frac{\frac{- 87}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = \frac{\frac{- 87}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.7

Move the negative in front of the fraction.

$y = \frac{- \frac{87}{5}}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.8

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{87}{5} \cdot \frac{1}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.9

Cancel the common factor of $3$ .

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Step 6.4.1.9.1

Move the leading negative in $- \frac{87}{5}$ into the numerator.

$y = \frac{- 87}{5} \cdot \frac{1}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.9.2

Factor $3$ out of $- 87$ .

$y = \frac{3 (- 29)}{5} \cdot \frac{1}{9}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.9.3

Factor $3$ out of $9$ .

$y = \frac{3 \cdot - 29}{5} \cdot \frac{1}{3 \cdot 3}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.9.4

Cancel the common factor.

$y = \frac{3 \cdot - 29}{5} \cdot \frac{1}{3 \cdot 3}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.9.5

Rewrite the expression.

$y = \frac{- 29}{5} \cdot \frac{1}{3}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = \frac{- 29}{5} \cdot \frac{1}{3}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.10

Multiply $\frac{- 29}{5}$ by $\frac{1}{3}$ .

$y = \frac{- 29}{5 \cdot 3}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.11

Simplify the expression.

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Step 6.4.1.11.1

Multiply $5$ by $3$ .

$y = \frac{- 29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 6.4.1.11.2

Move the negative in front of the fraction.

$y = - \frac{29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

$y = - \frac{29}{15}$

$x = \frac{221}{75}$

$z = - \frac{22}{25}$

Step 7

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

$(\frac{221}{75}, - \frac{29}{15}, - \frac{22}{25})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{221}{75}, - \frac{29}{15}, - \frac{22}{25})$

Equation Form:

$x = \frac{221}{75}, y = - \frac{29}{15}, z = - \frac{22}{25}$