Finite Math Examples

$10 x + 10 z = 100$ , $5 y - 5 z = 50$ , $10 x - 5 y = 50$

Step 1

Solve for $x$ in $10 x + 10 z = 100$ .

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Step 1.1

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

$10 x = 100 - 10 z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2

Divide each term in $10 x = 100 - 10 z$ by $10$ and simplify.

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Step 1.2.1

Divide each term in $10 x = 100 - 10 z$ by $10$ .

$\frac{10 x}{10} = \frac{100}{10} + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $10$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{100}{10} + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{100}{10} + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = \frac{100}{10} + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = \frac{100}{10} + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

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

Divide $100$ by $10$ .

$x = 10 + \frac{- 10 z}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

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

$x = 10 + \frac{10 (- z)}{10}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $10$ out of $10$ .

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

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 10 + \frac{10 (- z)}{10 \cdot 1}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 10 + \frac{- z}{1}$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 1.2.3.1.2.2.4

Divide $- z$ by $1$ .

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$10 x - 5 y = 50$

Step 2

Replace all occurrences of $x$ with $10 - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $10 x - 5 y = 50$ with $10 - z$ .

$10 (10 - z) - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$10 \cdot 10 + 10 (- z) - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

Step 2.2.1.2

Multiply $10$ by $10$ .

$100 + 10 (- z) - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

Step 2.2.1.3

Multiply $- 1$ by $10$ .

$100 - 10 z - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$100 - 10 z - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$100 - 10 z - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

$100 - 10 z - 5 y = 50$

$x = 10 - z$

$5 y - 5 z = 50$

Step 3

Reorder $10$ and $- z$ .

$x = - z + 10$

$100 - 10 z - 5 y = 50$

$5 y - 5 z = 50$

Step 4

Solve for $z$ in $100 - 10 z - 5 y = 50$ .

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Step 4.1

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

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Step 4.1.1

Subtract $100$ from both sides of the equation.

$- 10 z - 5 y = 50 - 100$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.1.2

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

$- 10 z = 50 - 100 + 5 y$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.1.3

Subtract $100$ from $50$ .

$- 10 z = - 50 + 5 y$

$x = - z + 10$

$5 y - 5 z = 50$

$- 10 z = - 50 + 5 y$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2

Divide each term in $- 10 z = - 50 + 5 y$ by $- 10$ and simplify.

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Step 4.2.1

Divide each term in $- 10 z = - 50 + 5 y$ by $- 10$ .

$\frac{- 10 z}{- 10} = \frac{- 50}{- 10} + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 10$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 10 z}{- 10} = \frac{- 50}{- 10} + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 50}{- 10} + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = \frac{- 50}{- 10} + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = \frac{- 50}{- 10} + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Divide $- 50$ by $- 10$ .

$z = 5 + \frac{5 y}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3.1.2

Cancel the common factor of $5$ and $- 10$ .

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Step 4.2.3.1.2.1

Factor $5$ out of $5 y$ .

$z = 5 + \frac{5 (y)}{- 10}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3.1.2.2

Cancel the common factors.

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Step 4.2.3.1.2.2.1

Factor $5$ out of $- 10$ .

$z = 5 + \frac{5 y}{5 \cdot - 2}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3.1.2.2.2

Cancel the common factor.

$z = 5 + \frac{5 y}{5 \cdot - 2}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3.1.2.2.3

Rewrite the expression.

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

$x = - z + 10$

$5 y - 5 z = 50$

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

$x = - z + 10$

$5 y - 5 z = 50$

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

$x = - z + 10$

$5 y - 5 z = 50$

Step 4.2.3.1.3

Move the negative in front of the fraction.

$z = 5 - \frac{y}{2}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = 5 - \frac{y}{2}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = 5 - \frac{y}{2}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = 5 - \frac{y}{2}$

$x = - z + 10$

$5 y - 5 z = 50$

$z = 5 - \frac{y}{2}$

$x = - z + 10$

$5 y - 5 z = 50$

Step 5

Replace all occurrences of $z$ with $5 - \frac{y}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $x = - z + 10$ with $5 - \frac{y}{2}$ .

$x = - (5 - \frac{y}{2}) + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- (5 - \frac{y}{2}) + 10$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$x = - 1 \cdot 5 + \frac{y}{2} + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.2.1.1.2

Multiply $- 1$ by $5$ .

$x = - 5 + \frac{y}{2} + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.2.1.1.3

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

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Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = - 5 + 1 (\frac{y}{2}) + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.2.1.1.3.2

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

$x = - 5 + \frac{y}{2} + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

$x = - 5 + \frac{y}{2} + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

$x = - 5 + \frac{y}{2} + 10$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.2.1.2

Add $- 5$ and $10$ .

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$5 y - 5 z = 50$

Step 5.3

Replace all occurrences of $z$ in $5 y - 5 z = 50$ with $5 - \frac{y}{2}$ .

$5 y - 5 (5 - \frac{y}{2}) = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $5 y - 5 (5 - \frac{y}{2})$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$5 y - 5 \cdot 5 - 5 (- \frac{y}{2}) = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.1.2

Multiply $- 5$ by $5$ .

$5 y - 25 - 5 (- \frac{y}{2}) = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.1.3

Multiply $- 5 (- \frac{y}{2})$ .

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Step 5.4.1.1.3.1

Multiply $- 1$ by $- 5$ .

$5 y - 25 + 5 (\frac{y}{2}) = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.1.3.2

Combine $5$ and $\frac{y}{2}$ .

$5 y - 25 + \frac{5 y}{2} = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$5 y - 25 + \frac{5 y}{2} = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$5 y - 25 + \frac{5 y}{2} = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.2

To write $5 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 y \cdot \frac{2}{2} + \frac{5 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.3

Simplify terms.

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Step 5.4.1.3.1

Combine $5 y$ and $\frac{2}{2}$ .

$\frac{5 y \cdot 2}{2} + \frac{5 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.3.2

Combine the numerators over the common denominator.

$\frac{5 y \cdot 2 + 5 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{5 y \cdot 2 + 5 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.4

Simplify the numerator.

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Step 5.4.1.4.1

Factor $5 y$ out of $5 y \cdot 2 + 5 y$ .

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Step 5.4.1.4.1.1

Factor $5 y$ out of $5 y \cdot 2$ .

$\frac{5 y (2) + 5 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.4.1.2

Factor $5 y$ out of $5 y$ .

$\frac{5 y (2) + 5 y (1)}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.4.1.3

Factor $5 y$ out of $5 y (2) + 5 y (1)$ .

$\frac{5 y (2 + 1)}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{5 y (2 + 1)}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.4.2

Add $2$ and $1$ .

$\frac{5 y \cdot 3}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 5.4.1.4.3

Multiply $3$ by $5$ .

$\frac{15 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{15 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{15 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{15 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{15 y}{2} - 25 = 50$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6

Solve for $y$ in $\frac{15 y}{2} - 25 = 50$ .

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Step 6.1

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

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Step 6.1.1

Add $25$ to both sides of the equation.

$\frac{15 y}{2} = 50 + 25$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.1.2

Add $50$ and $25$ .

$\frac{15 y}{2} = 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{15 y}{2} = 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.2

Multiply both sides of the equation by $\frac{2}{15}$ .

$\frac{2}{15} \cdot \frac{15 y}{2} = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{2}{15} \cdot \frac{15 y}{2}$ .

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Step 6.3.1.1.1

Cancel the common factor of $2$ .

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Step 6.3.1.1.1.1

Cancel the common factor.

$\frac{2}{15} \cdot \frac{15 y}{2} = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.1.1.1.2

Rewrite the expression.

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.1.1.2

Cancel the common factor of $15$ .

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Step 6.3.1.1.2.1

Factor $15$ out of $15 y$ .

$\frac{1}{15} \cdot (15 (y)) = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.1.1.2.2

Cancel the common factor.

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = \frac{2}{15} \cdot 75$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $\frac{2}{15} \cdot 75$ .

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Step 6.3.2.1.1

Cancel the common factor of $15$ .

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Step 6.3.2.1.1.1

Factor $15$ out of $75$ .

$y = \frac{2}{15} \cdot (15 (5))$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.2.1.1.2

Cancel the common factor.

$y = \frac{2}{15} \cdot (15 \cdot 5)$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.2.1.1.3

Rewrite the expression.

$y = 2 \cdot 5$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = 2 \cdot 5$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 6.3.2.1.2

Multiply $2$ by $5$ .

$y = 10$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = 10$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = 10$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = 10$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

$y = 10$

$x = \frac{y}{2} + 5$

$z = 5 - \frac{y}{2}$

Step 7

Replace all occurrences of $y$ with $10$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = \frac{y}{2} + 5$ with $10$ .

$x = \frac{10}{2} + 5$

$y = 10$

$z = 5 - \frac{y}{2}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{10}{2} + 5$ .

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Step 7.2.1.1

Divide $10$ by $2$ .

$x = 5 + 5$

$y = 10$

$z = 5 - \frac{y}{2}$

Step 7.2.1.2

Add $5$ and $5$ .

$x = 10$

$y = 10$

$z = 5 - \frac{y}{2}$

$x = 10$

$y = 10$

$z = 5 - \frac{y}{2}$

$x = 10$

$y = 10$

$z = 5 - \frac{y}{2}$

Step 7.3

Replace all occurrences of $y$ in $z = 5 - \frac{y}{2}$ with $10$ .

$z = 5 - \frac{10}{2}$

$x = 10$

$y = 10$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $5 - \frac{10}{2}$ .

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Step 7.4.1.1

Simplify each term.

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Step 7.4.1.1.1

Divide $10$ by $2$ .

$z = 5 - 1 \cdot 5$

$x = 10$

$y = 10$

Step 7.4.1.1.2

Multiply $- 1$ by $5$ .

$z = 5 - 5$

$x = 10$

$y = 10$

$z = 5 - 5$

$x = 10$

$y = 10$

Step 7.4.1.2

Subtract $5$ from $5$ .

$z = 0$

$x = 10$

$y = 10$

$z = 0$

$x = 10$

$y = 10$

$z = 0$

$x = 10$

$y = 10$

$z = 0$

$x = 10$

$y = 10$

Step 8

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

$(10, 10, 0)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(10, 10, 0)$

Equation Form:

$x = 10, y = 10, z = 0$