Finite Math Examples

$- 6 x - 10 y = 7$ , $- 10 x - 3 y = 1$

Step 1

Solve for $x$ in $- 6 x - 10 y = 7$ .

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Step 1.1

Add $10 y$ to both sides of the equation.

$- 6 x = 7 + 10 y$

$- 10 x - 3 y = 1$

Step 1.2

Divide each term in $- 6 x = 7 + 10 y$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 x = 7 + 10 y$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{7}{- 6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{7}{- 6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{- 6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

$x = \frac{7}{- 6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

$x = \frac{7}{- 6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

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 in front of the fraction.

$x = - \frac{7}{6} + \frac{10 y}{- 6}$

$- 10 x - 3 y = 1$

Step 1.2.3.1.2

Cancel the common factor of $10$ and $- 6$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $10 y$ .

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

$- 10 x - 3 y = 1$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $- 6$ .

$x = - \frac{7}{6} + \frac{2 (5 y)}{2 (- 3)}$

$- 10 x - 3 y = 1$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{7}{6} + \frac{2 (5 y)}{2 \cdot - 3}$

$- 10 x - 3 y = 1$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{7}{6} + \frac{5 y}{- 3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} + \frac{5 y}{- 3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} + \frac{5 y}{- 3}$

$- 10 x - 3 y = 1$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 10 x - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 10 x - 3 y = 1$

Step 2

Replace all occurrences of $x$ with $- \frac{7}{6} - \frac{5 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 10 x - 3 y = 1$ with $- \frac{7}{6} - \frac{5 y}{3}$ .

$- 10 (- \frac{7}{6} - \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 10 (- \frac{7}{6} - \frac{5 y}{3}) - 3 y$ .

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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.

$- 10 (- \frac{7}{6}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{7}{6}$ into the numerator.

$- 10 (\frac{- 7}{6}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.2.2

Factor $2$ out of $- 10$ .

$2 (- 5) (\frac{- 7}{6}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.2.3

Factor $2$ out of $6$ .

$2 \cdot (- 5 \frac{- 7}{2 \cdot 3}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.2.4

Cancel the common factor.

$2 \cdot (- 5 \frac{- 7}{2 \cdot 3}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.2.5

Rewrite the expression.

$- 5 (\frac{- 7}{3}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$- 5 (\frac{- 7}{3}) - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.3

Combine $- 5$ and $\frac{- 7}{3}$ .

$\frac{- 5 \cdot - 7}{3} - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.4

Multiply $- 5$ by $- 7$ .

$\frac{35}{3} - 10 (- \frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.5

Multiply $- 10 (- \frac{5 y}{3})$ .

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Step 2.2.1.1.5.1

Multiply $- 1$ by $- 10$ .

$\frac{35}{3} + 10 (\frac{5 y}{3}) - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.5.2

Combine $10$ and $\frac{5 y}{3}$ .

$\frac{35}{3} + \frac{10 (5 y)}{3} - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.1.5.3

Multiply $5$ by $10$ .

$\frac{35}{3} + \frac{50 y}{3} - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$\frac{35}{3} + \frac{50 y}{3} - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$\frac{35}{3} + \frac{50 y}{3} - 3 y = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.2

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

$\frac{35}{3} + \frac{50 y}{3} - 3 y \cdot \frac{3}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.3

Combine $- 3 y$ and $\frac{3}{3}$ .

$\frac{35}{3} + \frac{50 y}{3} + \frac{- 3 y \cdot 3}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{35}{3} + \frac{50 y - 3 y \cdot 3}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{35 + 50 y - 3 y \cdot 3}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.6

Multiply $3$ by $- 3$ .

$\frac{35 + 50 y - 9 y}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 2.2.1.7

Subtract $9 y$ from $50 y$ .

$\frac{35 + 41 y}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$\frac{35 + 41 y}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$\frac{35 + 41 y}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$\frac{35 + 41 y}{3} = 1$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3

Solve for $y$ in $\frac{35 + 41 y}{3} = 1$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{35 + 41 y}{3} \cdot 3 = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{35 + 41 y}{3} \cdot 3$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{35 + 41 y}{3} \cdot 3 = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.2.1.1.1.2

Rewrite the expression.

$35 + 41 y = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$35 + 41 y = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.2.1.1.2

Reorder $35$ and $41 y$ .

$41 y + 35 = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$41 y + 35 = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$41 y + 35 = 1 \cdot 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $3$ by $1$ .

$41 y + 35 = 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$41 y + 35 = 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$41 y + 35 = 3$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $35$ from both sides of the equation.

$41 y = 3 - 35$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3.1.2

Subtract $35$ from $3$ .

$41 y = - 32$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$41 y = - 32$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3.2

Divide each term in $41 y = - 32$ by $41$ and simplify.

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Step 3.3.2.1

Divide each term in $41 y = - 32$ by $41$ .

$\frac{41 y}{41} = \frac{- 32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $41$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{41 y}{41} = \frac{- 32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = \frac{- 32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = \frac{- 32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{5 y}{3}$

Step 4

Replace all occurrences of $y$ with $- \frac{32}{41}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{7}{6} - \frac{5 y}{3}$ with $- \frac{32}{41}$ .

$x = - \frac{7}{6} - \frac{5 (- \frac{32}{41})}{3}$

$y = - \frac{32}{41}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{7}{6} - \frac{5 (- \frac{32}{41})}{3}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1

Multiply $- 1$ by $5$ .

$x = - \frac{7}{6} - \frac{- 5 (\frac{32}{41})}{3}$

$y = - \frac{32}{41}$

Step 4.2.1.1.1.2

Combine $- 5$ and $\frac{32}{41}$ .

$x = - \frac{7}{6} - \frac{\frac{- 5 \cdot 32}{41}}{3}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} - \frac{\frac{- 5 \cdot 32}{41}}{3}$

$y = - \frac{32}{41}$

Step 4.2.1.1.2

Multiply $- 5$ by $32$ .

$x = - \frac{7}{6} - \frac{\frac{- 160}{41}}{3}$

$y = - \frac{32}{41}$

Step 4.2.1.1.3

Move the negative in front of the fraction.

$x = - \frac{7}{6} - \frac{- \frac{160}{41}}{3}$

$y = - \frac{32}{41}$

Step 4.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{7}{6} - (- \frac{160}{41} \cdot \frac{1}{3})$

$y = - \frac{32}{41}$

Step 4.2.1.1.5

Multiply $- \frac{160}{41} \cdot \frac{1}{3}$ .

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Step 4.2.1.1.5.1

Multiply $\frac{1}{3}$ by $\frac{160}{41}$ .

$x = - \frac{7}{6} + \frac{160}{3 \cdot 41}$

$y = - \frac{32}{41}$

Step 4.2.1.1.5.2

Multiply $3$ by $41$ .

$x = - \frac{7}{6} + \frac{160}{123}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} + \frac{160}{123}$

$y = - \frac{32}{41}$

Step 4.2.1.1.6

Multiply $- - \frac{160}{123}$ .

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Step 4.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$x = - \frac{7}{6} + 1 (\frac{160}{123})$

$y = - \frac{32}{41}$

Step 4.2.1.1.6.2

Multiply $\frac{160}{123}$ by $1$ .

$x = - \frac{7}{6} + \frac{160}{123}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} + \frac{160}{123}$

$y = - \frac{32}{41}$

$x = - \frac{7}{6} + \frac{160}{123}$

$y = - \frac{32}{41}$

Step 4.2.1.2

To write $- \frac{7}{6}$ as a fraction with a common denominator, multiply by $\frac{41}{41}$ .

$x = - \frac{7}{6} \cdot \frac{41}{41} + \frac{160}{123}$

$y = - \frac{32}{41}$

Step 4.2.1.3

To write $\frac{160}{123}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{7}{6} \cdot \frac{41}{41} + \frac{160}{123} \cdot \frac{2}{2}$

$y = - \frac{32}{41}$

Step 4.2.1.4

Write each expression with a common denominator of $246$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.4.1

Multiply $\frac{7}{6}$ by $\frac{41}{41}$ .

$x = - \frac{7 \cdot 41}{6 \cdot 41} + \frac{160}{123} \cdot \frac{2}{2}$

$y = - \frac{32}{41}$

Step 4.2.1.4.2

Multiply $6$ by $41$ .

$x = - \frac{7 \cdot 41}{246} + \frac{160}{123} \cdot \frac{2}{2}$

$y = - \frac{32}{41}$

Step 4.2.1.4.3

Multiply $\frac{160}{123}$ by $\frac{2}{2}$ .

$x = - \frac{7 \cdot 41}{246} + \frac{160 \cdot 2}{123 \cdot 2}$

$y = - \frac{32}{41}$

Step 4.2.1.4.4

Multiply $123$ by $2$ .

$x = - \frac{7 \cdot 41}{246} + \frac{160 \cdot 2}{246}$

$y = - \frac{32}{41}$

$x = - \frac{7 \cdot 41}{246} + \frac{160 \cdot 2}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{- 7 \cdot 41 + 160 \cdot 2}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 7$ by $41$ .

$x = \frac{- 287 + 160 \cdot 2}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.6.2

Multiply $160$ by $2$ .

$x = \frac{- 287 + 320}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.6.3

Add $- 287$ and $320$ .

$x = \frac{33}{246}$

$y = - \frac{32}{41}$

$x = \frac{33}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.7

Cancel the common factor of $33$ and $246$ .

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Step 4.2.1.7.1

Factor $3$ out of $33$ .

$x = \frac{3 (11)}{246}$

$y = - \frac{32}{41}$

Step 4.2.1.7.2

Cancel the common factors.

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Step 4.2.1.7.2.1

Factor $3$ out of $246$ .

$x = \frac{3 \cdot 11}{3 \cdot 82}$

$y = - \frac{32}{41}$

Step 4.2.1.7.2.2

Cancel the common factor.

$x = \frac{3 \cdot 11}{3 \cdot 82}$

$y = - \frac{32}{41}$

Step 4.2.1.7.2.3

Rewrite the expression.

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

$x = \frac{11}{82}$

$y = - \frac{32}{41}$

Step 5

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

$(\frac{11}{82}, - \frac{32}{41})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{11}{82}, - \frac{32}{41})$

Equation Form:

$x = \frac{11}{82}, y = - \frac{32}{41}$

Step 7