Finite Math Examples

$- \frac{2}{3} x + \frac{y}{2} = \frac{1}{6}$ , $\frac{x}{6} - y = \frac{5}{6}$

Step 1

Solve for $y$ in $\frac{x}{6} - y = \frac{5}{6}$ .

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Step 1.1

Subtract $\frac{x}{6}$ from both sides of the equation.

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2

Divide each term in $- y = \frac{5}{6} - \frac{x}{6}$ by $- 1$ and simplify.

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Step 1.2.1

Divide each term in $- y = \frac{5}{6} - \frac{x}{6}$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\frac{5}{6}}{- 1} + \frac{- \frac{x}{6}}{- 1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\frac{5}{6}}{- 1} + \frac{- \frac{x}{6}}{- 1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2.2.2

Divide $y$ by $1$ .

$y = \frac{\frac{5}{6}}{- 1} + \frac{- \frac{x}{6}}{- 1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

$y = \frac{\frac{5}{6}}{- 1} + \frac{- \frac{x}{6}}{- 1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

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

Move the negative one from the denominator of $\frac{\frac{5}{6}}{- 1}$ .

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2.3.1.2

Rewrite $- 1 \cdot \frac{5}{6}$ as $- \frac{5}{6}$ .

$y = - \frac{5}{6} + \frac{- \frac{x}{6}}{- 1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{5}{6} + \frac{\frac{x}{6}}{1}$

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 1.2.3.1.4

Divide $\frac{x}{6}$ by $1$ .

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

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

$- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$

Step 2

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

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Step 2.1

Replace all occurrences of $y$ in $- \frac{x \cdot 2}{3} + \frac{y}{2} = \frac{1}{6}$ with $- \frac{5}{6} + \frac{x}{6}$ .

$- \frac{x \cdot 2}{3} + \frac{- \frac{5}{6} + \frac{x}{6}}{2} = \frac{1}{6}$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- \frac{x \cdot 2}{3} + \frac{- \frac{5}{6} + \frac{x}{6}}{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Move $2$ to the left of $x$ .

$- \frac{2 \cdot x}{3} + \frac{- \frac{5}{6} + \frac{x}{6}}{2} = \frac{1}{6}$

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

Step 2.2.1.1.2

Combine the numerators over the common denominator.

$- \frac{2 x}{3} + \frac{\frac{- 5 + x}{6}}{2} = \frac{1}{6}$

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

Step 2.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2 x}{3} + \frac{- 5 + x}{6} \cdot \frac{1}{2} = \frac{1}{6}$

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

Step 2.2.1.1.4

Multiply $\frac{- 5 + x}{6} \cdot \frac{1}{2}$ .

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Step 2.2.1.1.4.1

Multiply $\frac{- 5 + x}{6}$ by $\frac{1}{2}$ .

$- \frac{2 x}{3} + \frac{- 5 + x}{6 \cdot 2} = \frac{1}{6}$

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

Step 2.2.1.1.4.2

Multiply $6$ by $2$ .

$- \frac{2 x}{3} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

$- \frac{2 x}{3} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

$- \frac{2 x}{3} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.2

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{2 x}{3} \cdot \frac{4}{4} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{2 x}{3}$ by $\frac{4}{4}$ .

$- \frac{2 x \cdot 4}{3 \cdot 4} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.3.2

Multiply $3$ by $4$ .

$- \frac{2 x \cdot 4}{12} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

$- \frac{2 x \cdot 4}{12} + \frac{- 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{- 2 x \cdot 4 - 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Multiply $4$ by $- 2$ .

$\frac{- 8 x - 5 + x}{12} = \frac{1}{6}$

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

Step 2.2.1.5.2

Add $- 8 x$ and $x$ .

$\frac{- 7 x - 5}{12} = \frac{1}{6}$

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

$\frac{- 7 x - 5}{12} = \frac{1}{6}$

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

Step 2.2.1.6

Simplify with factoring out.

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Step 2.2.1.6.1

Factor $- 1$ out of $- 7 x$ .

$\frac{- (7 x) - 5}{12} = \frac{1}{6}$

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

Step 2.2.1.6.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (7 x) - 1 \cdot 5}{12} = \frac{1}{6}$

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

Step 2.2.1.6.3

Factor $- 1$ out of $- (7 x) - 1 (5)$ .

$\frac{- (7 x + 5)}{12} = \frac{1}{6}$

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

Step 2.2.1.6.4

Simplify the expression.

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Step 2.2.1.6.4.1

Rewrite $- (7 x + 5)$ as $- 1 (7 x + 5)$ .

$\frac{- 1 (7 x + 5)}{12} = \frac{1}{6}$

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

Step 2.2.1.6.4.2

Move the negative in front of the fraction.

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

$- \frac{7 x + 5}{12} = \frac{1}{6}$

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

Step 3

Solve for $x$ in $- \frac{7 x + 5}{12} = \frac{1}{6}$ .

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Step 3.1

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{7 x + 5}{12}) = - 12 (\frac{1}{6})$

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

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

Simplify $- 12 (- \frac{7 x + 5}{12})$ .

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Step 3.2.1.1.1

Cancel the common factor of $12$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{7 x + 5}{12}$ into the numerator.

$- 12 \frac{- (7 x + 5)}{12} = - 12 (\frac{1}{6})$

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

Step 3.2.1.1.1.2

Factor $12$ out of $- 12$ .

$12 (- 1) (\frac{- (7 x + 5)}{12}) = - 12 (\frac{1}{6})$

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

Step 3.2.1.1.1.3

Cancel the common factor.

$12 \cdot (- 1 \frac{- (7 x + 5)}{12}) = - 12 (\frac{1}{6})$

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

Step 3.2.1.1.1.4

Rewrite the expression.

$7 x + 5 = - 12 (\frac{1}{6})$

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

$7 x + 5 = - 12 (\frac{1}{6})$

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

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (7 x + 5) = - 12 (\frac{1}{6})$

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

Step 3.2.1.1.2.2

Multiply $7 x + 5$ by $1$ .

$7 x + 5 = - 12 (\frac{1}{6})$

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

$7 x + 5 = - 12 (\frac{1}{6})$

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

$7 x + 5 = - 12 (\frac{1}{6})$

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

$7 x + 5 = - 12 (\frac{1}{6})$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Factor $6$ out of $- 12$ .

$7 x + 5 = 6 (- 2) (\frac{1}{6})$

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

Step 3.2.2.1.2

Cancel the common factor.

$7 x + 5 = 6 \cdot (- 2 (\frac{1}{6}))$

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

Step 3.2.2.1.3

Rewrite the expression.

$7 x + 5 = - 2$

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

$7 x + 5 = - 2$

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

$7 x + 5 = - 2$

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

$7 x + 5 = - 2$

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

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 $5$ from both sides of the equation.

$7 x = - 2 - 5$

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

Step 3.3.2

Subtract $5$ from $- 2$ .

$7 x = - 7$

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

$7 x = - 7$

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

Step 3.4

Divide each term in $7 x = - 7$ by $7$ and simplify.

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Step 3.4.1

Divide each term in $7 x = - 7$ by $7$ .

$\frac{7 x}{7} = \frac{- 7}{7}$

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 7}{7}$

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

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{7}$

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

$x = \frac{- 7}{7}$

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

$x = \frac{- 7}{7}$

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 7$ by $7$ .

$x = - 1$

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

$x = - 1$

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

$x = - 1$

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

$x = - 1$

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

Step 4

Replace all occurrences of $x$ with $- 1$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = - \frac{5}{6} + \frac{x}{6}$ with $- 1$ .

$y = - \frac{5}{6} + \frac{- 1}{6}$

$x = - 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{6} + \frac{- 1}{6}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{- 5 - 1}{6}$

$x = - 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Subtract $1$ from $- 5$ .

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

$x = - 1$

Step 4.2.1.2.2

Divide $- 6$ by $6$ .

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

$y = - 1$

$x = - 1$

Step 5

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

$(- 1, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 1, - 1)$

Equation Form:

$x = - 1, y = - 1$

Step 7