Finite Math Examples

$\frac{x}{5} + y = \frac{6}{5}$ , $\frac{x}{10} + \frac{y}{3} = \frac{5}{6}$

Step 1

Subtract $\frac{x}{5}$ from both sides of the equation.

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x}{10} + \frac{y}{3} = \frac{5}{6}$

Step 2

Replace all occurrences of $y$ with $\frac{6}{5} - \frac{x}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $\frac{x}{10} + \frac{y}{3} = \frac{5}{6}$ with $\frac{6}{5} - \frac{x}{5}$ .

$\frac{x}{10} + \frac{\frac{6}{5} - \frac{x}{5}}{3} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\frac{x}{10} + \frac{\frac{6}{5} - \frac{x}{5}}{3}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Combine the numerators over the common denominator.

$\frac{x}{10} + \frac{\frac{6 - x}{5}}{3} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{10} + \frac{6 - x}{5} \cdot \frac{1}{3} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.1.3

Multiply $\frac{6 - x}{5} \cdot \frac{1}{3}$ .

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Step 2.2.1.1.3.1

Multiply $\frac{6 - x}{5}$ by $\frac{1}{3}$ .

$\frac{x}{10} + \frac{6 - x}{5 \cdot 3} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.1.3.2

Multiply $5$ by $3$ .

$\frac{x}{10} + \frac{6 - x}{15} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x}{10} + \frac{6 - x}{15} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x}{10} + \frac{6 - x}{15} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.2

To write $\frac{x}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{10} \cdot \frac{3}{3} + \frac{6 - x}{15} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.3

To write $\frac{6 - x}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{10} \cdot \frac{3}{3} + \frac{6 - x}{15} \cdot \frac{2}{2} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.4

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.4.1

Multiply $\frac{x}{10}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{10 \cdot 3} + \frac{6 - x}{15} \cdot \frac{2}{2} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.4.2

Multiply $10$ by $3$ .

$\frac{x \cdot 3}{30} + \frac{6 - x}{15} \cdot \frac{2}{2} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.4.3

Multiply $\frac{6 - x}{15}$ by $\frac{2}{2}$ .

$\frac{x \cdot 3}{30} + \frac{(6 - x) \cdot 2}{15 \cdot 2} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.4.4

Multiply $15$ by $2$ .

$\frac{x \cdot 3}{30} + \frac{(6 - x) \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x \cdot 3}{30} + \frac{(6 - x) \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + (6 - x) \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.6

Simplify the numerator.

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Step 2.2.1.6.1

Move $3$ to the left of $x$ .

$\frac{3 \cdot x + (6 - x) \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.6.2

Apply the distributive property.

$\frac{3 x + 6 \cdot 2 - x \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.6.3

Multiply $6$ by $2$ .

$\frac{3 x + 12 - x \cdot 2}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.6.4

Multiply $2$ by $- 1$ .

$\frac{3 x + 12 - 2 x}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 2.2.1.6.5

Subtract $2 x$ from $3 x$ .

$\frac{x + 12}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x + 12}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x + 12}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x + 12}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

$\frac{x + 12}{30} = \frac{5}{6}$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3

Solve for $x$ in $\frac{x + 12}{30} = \frac{5}{6}$ .

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Step 3.1

Multiply both sides of the equation by $30$ .

$30 (\frac{x + 12}{30}) = 30 (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $30$ .

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Step 3.2.1.1.1

Cancel the common factor.

$30 (\frac{x + 12}{30}) = 30 (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2.1.1.2

Rewrite the expression.

$x + 12 = 30 (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 30 (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 30 (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $30 (\frac{5}{6})$ .

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Step 3.2.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1.1

Factor $6$ out of $30$ .

$x + 12 = 6 (5) (\frac{5}{6})$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2.2.1.1.2

Cancel the common factor.

$x + 12 = 6 \cdot (5 (\frac{5}{6}))$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2.2.1.1.3

Rewrite the expression.

$x + 12 = 5 \cdot 5$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 5 \cdot 5$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.2.2.1.2

Multiply $5$ by $5$ .

$x + 12 = 25$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 25$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 25$

$y = \frac{6}{5} - \frac{x}{5}$

$x + 12 = 25$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $12$ from both sides of the equation.

$x = 25 - 12$

$y = \frac{6}{5} - \frac{x}{5}$

Step 3.3.2

Subtract $12$ from $25$ .

$x = 13$

$y = \frac{6}{5} - \frac{x}{5}$

$x = 13$

$y = \frac{6}{5} - \frac{x}{5}$

$x = 13$

$y = \frac{6}{5} - \frac{x}{5}$

Step 4

Replace all occurrences of $x$ with $13$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{6}{5} - \frac{x}{5}$ with $13$ .

$y = \frac{6}{5} - \frac{13}{5}$

$x = 13$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{6}{5} - \frac{13}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{6 - 13}{5}$

$x = 13$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Subtract $13$ from $6$ .

$y = \frac{- 7}{5}$

$x = 13$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$y = - \frac{7}{5}$

$x = 13$

$y = - \frac{7}{5}$

$x = 13$

$y = - \frac{7}{5}$

$x = 13$

$y = - \frac{7}{5}$

$x = 13$

$y = - \frac{7}{5}$

$x = 13$

Step 5

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

$(13, - \frac{7}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(13, - \frac{7}{5})$

Equation Form:

$x = 13, y = - \frac{7}{5}$

Step 7