Finite Math Examples

$z = 7 x + 11 y$ , $0.5 x + 0.25 y = 117$ , $0.5 x + 0.75 y = 207$

Step 1

Solve for $x$ in $0.5 x + 0.25 y = 117$ .

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Step 1.1

Subtract $0.25 y$ from both sides of the equation.

$0.5 x = 117 - 0.25 y$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2

Divide each term in $0.5 x = 117 - 0.25 y$ by $0.5$ and simplify.

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Step 1.2.1

Divide each term in $0.5 x = 117 - 0.25 y$ by $0.5$ .

$\frac{0.5 x}{0.5} = \frac{117}{0.5} + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $0.5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{0.5 x}{0.5} = \frac{117}{0.5} + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{117}{0.5} + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = \frac{117}{0.5} + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = \frac{117}{0.5} + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

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 $117$ by $0.5$ .

$x = 234 + \frac{- 0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = 234 - \frac{0.25 y}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.3

Factor $0.25$ out of $0.25 y$ .

$x = 234 - \frac{0.25 (y)}{0.5}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.4

Factor $0.5$ out of $0.5$ .

$x = 234 - \frac{0.25 (y)}{0.5 (1)}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.5

Separate fractions.

$x = 234 - (\frac{0.25}{0.5} \cdot \frac{y}{1})$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.6

Divide $0.25$ by $0.5$ .

$x = 234 - (0.5 (\frac{y}{1}))$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.7

Cancel the common factor of $0.5$ .

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Step 1.2.3.1.7.1

Factor $0.5$ out of $1$ .

$x = 234 - (0.5 (\frac{y}{0.5 \cdot 2}))$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.7.2

Cancel the common factor.

$x = 234 - (0.5 (\frac{y}{0.5 \cdot 2}))$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 1.2.3.1.7.3

Rewrite the expression.

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

$x = 234 - \frac{y}{2}$

$z = 7 x + 11 y$

$0.5 x + 0.75 y = 207$

Step 2

Replace all occurrences of $x$ with $234 - \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $z = 7 x + 11 y$ with $234 - \frac{y}{2}$ .

$z = 7 (234 - \frac{y}{2}) + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $7 (234 - \frac{y}{2}) + 11 y$ .

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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.

$z = 7 \cdot 234 + 7 (- \frac{y}{2}) + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.1.2

Multiply $7$ by $234$ .

$z = 1638 + 7 (- \frac{y}{2}) + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.1.3

Multiply $7 (- \frac{y}{2})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $7$ .

$z = 1638 - 7 \frac{y}{2} + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.1.3.2

Combine $- 7$ and $\frac{y}{2}$ .

$z = 1638 + \frac{- 7 y}{2} + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{- 7 y}{2} + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$z = 1638 - \frac{7 y}{2} + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 - \frac{7 y}{2} + 11 y$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.2

To write $11 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z = 1638 - \frac{7 y}{2} + 11 y \cdot \frac{2}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $11 y$ and $\frac{2}{2}$ .

$z = 1638 - \frac{7 y}{2} + \frac{11 y \cdot 2}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$z = 1638 + \frac{- 7 y + 11 y \cdot 2}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{- 7 y + 11 y \cdot 2}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $- 7 y + 11 y \cdot 2$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $- 7 y$ .

$z = 1638 + \frac{y \cdot - 7 + 11 y \cdot 2}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4.1.1.2

Factor $y$ out of $11 y \cdot 2$ .

$z = 1638 + \frac{y \cdot - 7 + y (11 \cdot 2)}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot - 7 + y (11 \cdot 2)$ .

$z = 1638 + \frac{y (- 7 + 11 \cdot 2)}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{y (- 7 + 11 \cdot 2)}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4.1.2

Multiply $11$ by $2$ .

$z = 1638 + \frac{y (- 7 + 22)}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4.1.3

Add $- 7$ and $22$ .

$z = 1638 + \frac{y \cdot 15}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{y \cdot 15}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.2.1.4.2

Move $15$ to the left of $y$ .

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$0.5 x + 0.75 y = 207$

Step 2.3

Replace all occurrences of $x$ in $0.5 x + 0.75 y = 207$ with $234 - \frac{y}{2}$ .

$0.5 (234 - \frac{y}{2}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $0.5 (234 - \frac{y}{2}) + 0.75 y$ .

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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.

$0.5 \cdot 234 + 0.5 (- \frac{y}{2}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.2

Multiply $0.5$ by $234$ .

$117 + 0.5 (- \frac{y}{2}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.3

Cancel the common factor of $0.5$ .

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Step 2.4.1.1.3.1

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

$117 + 0.5 (\frac{- y}{2}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.3.2

Factor $0.5$ out of $2$ .

$117 + 0.5 (\frac{- y}{0.5 (4)}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.3.3

Cancel the common factor.

$117 + 0.5 (\frac{- y}{0.5 \cdot 4}) + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.3.4

Rewrite the expression.

$117 + \frac{- y}{4} + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{- y}{4} + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.1.4

Move the negative in front of the fraction.

$117 - \frac{y}{4} + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 - \frac{y}{4} + 0.75 y = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.2

To write $0.75 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$117 - \frac{y}{4} + 0.75 y \cdot \frac{4}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.3

Simplify terms.

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Step 2.4.1.3.1

Combine $0.75 y$ and $\frac{4}{4}$ .

$117 - \frac{y}{4} + \frac{0.75 y \cdot 4}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.3.2

Combine the numerators over the common denominator.

$117 + \frac{- y + 0.75 y \cdot 4}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{- y + 0.75 y \cdot 4}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4

Simplify each term.

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Step 2.4.1.4.1

Simplify the numerator.

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Step 2.4.1.4.1.1

Factor $0.25 y$ out of $- y + 0.75 y \cdot 4$ .

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Step 2.4.1.4.1.1.1

Move $y$ .

$117 + \frac{- y + 0.75 \cdot (4 y)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.1.1.2

Factor $0.25 y$ out of $- y$ .

$117 + \frac{0.25 y (- 4) + 0.75 \cdot (4 y)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.1.1.3

Factor $0.25 y$ out of $0.75 \cdot 4 y$ .

$117 + \frac{0.25 y (- 4) + 0.25 y (3 \cdot 4)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.1.1.4

Factor $0.25 y$ out of $0.25 y (- 4) + 0.25 y (3 \cdot 4)$ .

$117 + \frac{0.25 y (- 4 + 3 \cdot 4)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{0.25 y (- 4 + 3 \cdot 4)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.1.2

Combine exponents.

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Step 2.4.1.4.1.2.1

Multiply $3$ by $4$ .

$117 + \frac{0.25 y (- 4 + 12)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.1.2.2

Multiply $- 4 + 12$ by $0.25$ .

$117 + \frac{2 y}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{2 y}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{2 y}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.2

Cancel the common factor of $2$ and $4$ .

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Step 2.4.1.4.2.1

Factor $2$ out of $2 y$ .

$117 + \frac{2 (y)}{4} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.2.2

Cancel the common factors.

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Step 2.4.1.4.2.2.1

Factor $2$ out of $4$ .

$117 + \frac{2 y}{2 \cdot 2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.2.2.2

Cancel the common factor.

$117 + \frac{2 y}{2 \cdot 2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 2.4.1.4.2.2.3

Rewrite the expression.

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$117 + \frac{y}{2} = 207$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3

Solve for $y$ in $117 + \frac{y}{2} = 207$ .

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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 $117$ from both sides of the equation.

$\frac{y}{2} = 207 - 117$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3.1.2

Subtract $117$ from $207$ .

$\frac{y}{2} = 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$\frac{y}{2} = 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 (\frac{y}{2}) = 2 \cdot 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1

Cancel the common factor.

$2 (\frac{y}{2}) = 2 \cdot 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3.3.1.1.2

Rewrite the expression.

$y = 2 \cdot 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$y = 2 \cdot 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$y = 2 \cdot 90$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $2$ by $90$ .

$y = 180$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$y = 180$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$y = 180$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

$y = 180$

$z = 1638 + \frac{15 y}{2}$

$x = 234 - \frac{y}{2}$

Step 4

Replace all occurrences of $y$ with $180$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $z = 1638 + \frac{15 y}{2}$ with $180$ .

$z = 1638 + \frac{15 (180)}{2}$

$y = 180$

$x = 234 - \frac{y}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $1638 + \frac{15 (180)}{2}$ .

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Step 4.2.1.1

Multiply $15$ by $180$ .

$z = 1638 + \frac{2700}{2}$

$y = 180$

$x = 234 - \frac{y}{2}$

Step 4.2.1.2

Divide $2700$ by $2$ .

$z = 1638 + 1350$

$y = 180$

$x = 234 - \frac{y}{2}$

Step 4.2.1.3

Add $1638$ and $1350$ .

$z = 2988$

$y = 180$

$x = 234 - \frac{y}{2}$

$z = 2988$

$y = 180$

$x = 234 - \frac{y}{2}$

$z = 2988$

$y = 180$

$x = 234 - \frac{y}{2}$

Step 4.3

Replace all occurrences of $y$ in $x = 234 - \frac{y}{2}$ with $180$ .

$x = 234 - \frac{180}{2}$

$z = 2988$

$y = 180$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $234 - \frac{180}{2}$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Divide $180$ by $2$ .

$x = 234 - 1 \cdot 90$

$z = 2988$

$y = 180$

Step 4.4.1.1.2

Multiply $- 1$ by $90$ .

$x = 234 - 90$

$z = 2988$

$y = 180$

$x = 234 - 90$

$z = 2988$

$y = 180$

Step 4.4.1.2

Subtract $90$ from $234$ .

$x = 144$

$z = 2988$

$y = 180$

$x = 144$

$z = 2988$

$y = 180$

$x = 144$

$z = 2988$

$y = 180$

$x = 144$

$z = 2988$

$y = 180$

Step 5

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

$(144, 180, 2988)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(144, 180, 2988)$

Equation Form:

$x = 144, y = 180, z = 2988$