Finite Math Examples

$- 10 x + 6 y = 6$ , $3 x + 3 y = 15$

Step 1

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

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Step 1.1

Subtract $6 y$ from both sides of the equation.

$- 10 x = 6 - 6 y$

$3 x + 3 y = 15$

Step 1.2

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

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Step 1.2.1

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

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

$3 x + 3 y = 15$

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.

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

$3 x + 3 y = 15$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

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

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Step 1.2.3.1.1.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{- 10} + \frac{- 6 y}{- 10}$

$3 x + 3 y = 15$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $- 10$ .

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

$3 x + 3 y = 15$

Step 1.2.3.1.1.2.2

Cancel the common factor.

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

$3 x + 3 y = 15$

Step 1.2.3.1.1.2.3

Rewrite the expression.

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$3 x + 3 y = 15$

Step 1.2.3.1.3

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

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Step 1.2.3.1.3.1

Factor $- 2$ out of $- 6 y$ .

$x = - \frac{3}{5} + \frac{- 2 (3 y)}{- 10}$

$3 x + 3 y = 15$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $- 2$ out of $- 10$ .

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

$3 x + 3 y = 15$

Step 1.2.3.1.3.2.2

Cancel the common factor.

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

$3 x + 3 y = 15$

Step 1.2.3.1.3.2.3

Rewrite the expression.

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

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

$3 x + 3 y = 15$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $3 x + 3 y = 15$ with $- \frac{3}{5} + \frac{3 y}{5}$ .

$3 (- \frac{3}{5} + \frac{3 y}{5}) + 3 y = 15$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{3}{5} + \frac{3 y}{5}) + 3 y$ .

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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.

$3 (- \frac{3}{5}) + 3 (\frac{3 y}{5}) + 3 y = 15$

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

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{3}{5}) + 3 (\frac{3 y}{5}) + 3 y = 15$

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

Step 2.2.1.1.2.2

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

$\frac{- 3 \cdot 3}{5} + 3 (\frac{3 y}{5}) + 3 y = 15$

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

Step 2.2.1.1.2.3

Multiply $- 3$ by $3$ .

$\frac{- 9}{5} + 3 (\frac{3 y}{5}) + 3 y = 15$

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

$\frac{- 9}{5} + 3 (\frac{3 y}{5}) + 3 y = 15$

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

Step 2.2.1.1.3

Multiply $3 \frac{3 y}{5}$ .

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Step 2.2.1.1.3.1

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

$\frac{- 9}{5} + \frac{3 (3 y)}{5} + 3 y = 15$

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

Step 2.2.1.1.3.2

Multiply $3$ by $3$ .

$\frac{- 9}{5} + \frac{9 y}{5} + 3 y = 15$

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

$\frac{- 9}{5} + \frac{9 y}{5} + 3 y = 15$

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{9}{5} + \frac{9 y}{5} + 3 y = 15$

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

$- \frac{9}{5} + \frac{9 y}{5} + 3 y = 15$

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

Step 2.2.1.2

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{9}{5} + \frac{9 y}{5} + 3 y \cdot \frac{5}{5} = 15$

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

Step 2.2.1.3

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

$- \frac{9}{5} + \frac{9 y}{5} + \frac{3 y \cdot 5}{5} = 15$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{9}{5} + \frac{9 y + 3 y \cdot 5}{5} = 15$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 9 + 9 y + 3 y \cdot 5}{5} = 15$

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

Step 2.2.1.6

Multiply $5$ by $3$ .

$\frac{- 9 + 9 y + 15 y}{5} = 15$

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

Step 2.2.1.7

Add $9 y$ and $15 y$ .

$\frac{- 9 + 24 y}{5} = 15$

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

Step 2.2.1.8

Factor $3$ out of $- 9 + 24 y$ .

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Step 2.2.1.8.1

Factor $3$ out of $- 9$ .

$\frac{3 (- 3) + 24 y}{5} = 15$

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

Step 2.2.1.8.2

Factor $3$ out of $24 y$ .

$\frac{3 (- 3) + 3 (8 y)}{5} = 15$

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

Step 2.2.1.8.3

Factor $3$ out of $3 (- 3) + 3 (8 y)$ .

$\frac{3 (- 3 + 8 y)}{5} = 15$

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

$\frac{3 (- 3 + 8 y)}{5} = 15$

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

Step 2.2.1.9

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{3 (- 1 \cdot 3 + 8 y)}{5} = 15$

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

Step 2.2.1.10

Factor $- 1$ out of $8 y$ .

$\frac{3 (- 1 \cdot 3 - (- 8 y))}{5} = 15$

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

Step 2.2.1.11

Factor $- 1$ out of $- 1 (3) - (- 8 y)$ .

$\frac{3 (- 1 (3 - 8 y))}{5} = 15$

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

Step 2.2.1.12

Move the negative in front of the fraction.

$- \frac{3 (3 - 8 y)}{5} = 15$

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

$- \frac{3 (3 - 8 y)}{5} = 15$

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

$- \frac{3 (3 - 8 y)}{5} = 15$

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

$- \frac{3 (3 - 8 y)}{5} = 15$

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

Step 3

Solve for $y$ in $- \frac{3 (3 - 8 y)}{5} = 15$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} \cdot (- \frac{3 (3 - 8 y)}{5}) = - \frac{5}{3} \cdot 15$

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

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

Simplify $- \frac{5}{3} (- \frac{3 (3 - 8 y)}{5})$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} \cdot (- \frac{3 (3 - 8 y)}{5}) = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.1.2

Move the leading negative in $- \frac{3 (3 - 8 y)}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- 3 (3 - 8 y)}{5} = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- 3 (3 - 8 y)}{5} = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- 3 (3 - 8 y)}{5} = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} \cdot (- 3 (3 - 8 y)) = - \frac{5}{3} \cdot 15$

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

$\frac{- 1}{3} \cdot (- 3 (3 - 8 y)) = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Factor $3$ out of $- 3 (3 - 8 y)$ .

$\frac{- 1}{3} \cdot (3 (- (3 - 8 y))) = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 (- (3 - 8 y))) = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.2.3

Rewrite the expression.

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (3 - 8 y) = - \frac{5}{3} \cdot 15$

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

Step 3.2.1.1.3.2

Multiply $3 - 8 y$ by $1$ .

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

$3 - 8 y = - \frac{5}{3} \cdot 15$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{5}{3} \cdot 15$ .

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$3 - 8 y = \frac{- 5}{3} \cdot 15$

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

Step 3.2.2.1.1.2

Factor $3$ out of $15$ .

$3 - 8 y = \frac{- 5}{3} \cdot (3 (5))$

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

Step 3.2.2.1.1.3

Cancel the common factor.

$3 - 8 y = \frac{- 5}{3} \cdot (3 \cdot 5)$

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

Step 3.2.2.1.1.4

Rewrite the expression.

$3 - 8 y = - 5 \cdot 5$

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

$3 - 8 y = - 5 \cdot 5$

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

Step 3.2.2.1.2

Multiply $- 5$ by $5$ .

$3 - 8 y = - 25$

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

$3 - 8 y = - 25$

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

$3 - 8 y = - 25$

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

$3 - 8 y = - 25$

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

Step 3.3

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

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Step 3.3.1

Subtract $3$ from both sides of the equation.

$- 8 y = - 25 - 3$

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

Step 3.3.2

Subtract $3$ from $- 25$ .

$- 8 y = - 28$

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

$- 8 y = - 28$

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

Step 3.4

Divide each term in $- 8 y = - 28$ by $- 8$ and simplify.

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Step 3.4.1

Divide each term in $- 8 y = - 28$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- 28}{- 8}$

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 8$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- 28}{- 8}$

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 28}{- 8}$

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

$y = \frac{- 28}{- 8}$

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

$y = \frac{- 28}{- 8}$

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $- 28$ and $- 8$ .

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Step 3.4.3.1.1

Factor $- 4$ out of $- 28$ .

$y = \frac{- 4 \cdot 7}{- 8}$

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

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $- 4$ out of $- 8$ .

$y = \frac{- 4 \cdot 7}{- 4 \cdot 2}$

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

Step 3.4.3.1.2.2

Cancel the common factor.

$y = \frac{- 4 \cdot 7}{- 4 \cdot 2}$

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

Step 3.4.3.1.2.3

Rewrite the expression.

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

$y = \frac{7}{2}$

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

Step 4

Replace all occurrences of $y$ with $\frac{7}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{3}{5} + \frac{3 y}{5}$ with $\frac{7}{2}$ .

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

$y = \frac{7}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3}{5} + \frac{3 (\frac{7}{2})}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = \frac{7}{2}$

Step 4.2.1.2

Multiply $3 (\frac{7}{2})$ .

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Step 4.2.1.2.1

Combine $3$ and $\frac{7}{2}$ .

$x = \frac{- 3 + \frac{3 \cdot 7}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.2.2

Multiply $3$ by $7$ .

$x = \frac{- 3 + \frac{21}{2}}{5}$

$y = \frac{7}{2}$

$x = \frac{- 3 + \frac{21}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.3

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

$x = \frac{- 3 \cdot \frac{2}{2} + \frac{21}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.4

Combine $- 3$ and $\frac{2}{2}$ .

$x = \frac{\frac{- 3 \cdot 2}{2} + \frac{21}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 3 \cdot 2 + 21}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 3$ by $2$ .

$x = \frac{\frac{- 6 + 21}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.6.2

Add $- 6$ and $21$ .

$x = \frac{\frac{15}{2}}{5}$

$y = \frac{7}{2}$

$x = \frac{\frac{15}{2}}{5}$

$y = \frac{7}{2}$

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{15}{2} \cdot \frac{1}{5}$

$y = \frac{7}{2}$

Step 4.2.1.8

Cancel the common factor of $5$ .

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Step 4.2.1.8.1

Factor $5$ out of $15$ .

$x = \frac{5 (3)}{2} \cdot \frac{1}{5}$

$y = \frac{7}{2}$

Step 4.2.1.8.2

Cancel the common factor.

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

$y = \frac{7}{2}$

Step 4.2.1.8.3

Rewrite the expression.

$x = \frac{3}{2}$

$y = \frac{7}{2}$

$x = \frac{3}{2}$

$y = \frac{7}{2}$

$x = \frac{3}{2}$

$y = \frac{7}{2}$

$x = \frac{3}{2}$

$y = \frac{7}{2}$

$x = \frac{3}{2}$

$y = \frac{7}{2}$

Step 5

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

$(\frac{3}{2}, \frac{7}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{3}{2}, \frac{7}{2})$

Equation Form:

$x = \frac{3}{2}, y = \frac{7}{2}$

Step 7