Finite Math Examples

$(x - y) - (6 x + 8 y) = - (10 x + 5 y + 3)$ , $(x + y) - (9 y - 11 x) = 2 y - 2 x$

Step 1

Solve for $x$ in $- 5 x - 9 y = - 10 x - 5 y - 3$ .

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Step 1.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1.1

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

$- 5 x - 9 y + 10 x = - 5 y - 3$

$12 x - 8 y = 2 y - 2 x$

Step 1.1.2

Add $- 5 x$ and $10 x$ .

$5 x - 9 y = - 5 y - 3$

$12 x - 8 y = 2 y - 2 x$

$5 x - 9 y = - 5 y - 3$

$12 x - 8 y = 2 y - 2 x$

Step 1.2

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

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Step 1.2.1

Add $9 y$ to both sides of the equation.

$5 x = - 5 y - 3 + 9 y$

$12 x - 8 y = 2 y - 2 x$

Step 1.2.2

Add $- 5 y$ and $9 y$ .

$5 x = 4 y - 3$

$12 x - 8 y = 2 y - 2 x$

$5 x = 4 y - 3$

$12 x - 8 y = 2 y - 2 x$

Step 1.3

Divide each term in $5 x = 4 y - 3$ by $5$ and simplify.

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Step 1.3.1

Divide each term in $5 x = 4 y - 3$ by $5$ .

$\frac{5 x}{5} = \frac{4 y}{5} + \frac{- 3}{5}$

$12 x - 8 y = 2 y - 2 x$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $5$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4 y}{5} + \frac{- 3}{5}$

$12 x - 8 y = 2 y - 2 x$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 y}{5} + \frac{- 3}{5}$

$12 x - 8 y = 2 y - 2 x$

$x = \frac{4 y}{5} + \frac{- 3}{5}$

$12 x - 8 y = 2 y - 2 x$

$x = \frac{4 y}{5} + \frac{- 3}{5}$

$12 x - 8 y = 2 y - 2 x$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative in front of the fraction.

$x = \frac{4 y}{5} - \frac{3}{5}$

$12 x - 8 y = 2 y - 2 x$

$x = \frac{4 y}{5} - \frac{3}{5}$

$12 x - 8 y = 2 y - 2 x$

$x = \frac{4 y}{5} - \frac{3}{5}$

$12 x - 8 y = 2 y - 2 x$

$x = \frac{4 y}{5} - \frac{3}{5}$

$12 x - 8 y = 2 y - 2 x$

Step 2

Replace all occurrences of $x$ with $\frac{4 y}{5} - \frac{3}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $12 x - 8 y = 2 y - 2 x$ with $\frac{4 y}{5} - \frac{3}{5}$ .

$12 (\frac{4 y}{5} - \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2

Simplify $12 (\frac{4 y}{5} - \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $12 (\frac{4 y}{5} - \frac{3}{5}) - 8 y$ .

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Step 2.2.1.1.1

Simplify each term.

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Step 2.2.1.1.1.1

Apply the distributive property.

$12 (\frac{4 y}{5}) + 12 (- \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.2

Multiply $12 \frac{4 y}{5}$ .

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Step 2.2.1.1.1.2.1

Combine $12$ and $\frac{4 y}{5}$ .

$\frac{12 (4 y)}{5} + 12 (- \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.2.2

Multiply $4$ by $12$ .

$\frac{48 y}{5} + 12 (- \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{48 y}{5} + 12 (- \frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.3

Multiply $12 (- \frac{3}{5})$ .

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Step 2.2.1.1.1.3.1

Multiply $- 1$ by $12$ .

$\frac{48 y}{5} - 12 (\frac{3}{5}) - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.3.2

Combine $- 12$ and $\frac{3}{5}$ .

$\frac{48 y}{5} + \frac{- 12 \cdot 3}{5} - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.3.3

Multiply $- 12$ by $3$ .

$\frac{48 y}{5} + \frac{- 36}{5} - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{48 y}{5} + \frac{- 36}{5} - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.1.4

Move the negative in front of the fraction.

$\frac{48 y}{5} - \frac{36}{5} - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{48 y}{5} - \frac{36}{5} - 8 y = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.2

To write $- 8 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{48 y}{5} - 8 y \cdot \frac{5}{5} - \frac{36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.3

Combine $- 8 y$ and $\frac{5}{5}$ .

$\frac{48 y}{5} + \frac{- 8 y \cdot 5}{5} - \frac{36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.4

Combine the numerators over the common denominator.

$\frac{48 y - 8 y \cdot 5}{5} - \frac{36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.5

Combine the numerators over the common denominator.

$\frac{48 y - 8 y \cdot 5 - 36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.6

Multiply $5$ by $- 8$ .

$\frac{48 y - 40 y - 36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.7

Subtract $40 y$ from $48 y$ .

$\frac{8 y - 36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.8

Factor $4$ out of $8 y - 36$ .

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Step 2.2.1.1.8.1

Factor $4$ out of $8 y$ .

$\frac{4 (2 y) - 36}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.8.2

Factor $4$ out of $- 36$ .

$\frac{4 (2 y) + 4 (- 9)}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.1.1.8.3

Factor $4$ out of $4 (2 y) + 4 (- 9)$ .

$\frac{4 (2 y - 9)}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $2 y - 2 (\frac{4 y}{5} - \frac{3}{5})$ .

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Step 2.2.2.1.1

Simplify each term.

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Step 2.2.2.1.1.1

Apply the distributive property.

$\frac{4 (2 y - 9)}{5} = 2 y - 2 \frac{4 y}{5} - 2 (- \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.2

Multiply $- 2 \frac{4 y}{5}$ .

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Step 2.2.2.1.1.2.1

Combine $- 2$ and $\frac{4 y}{5}$ .

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 2 (4 y)}{5} - 2 (- \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.2.2

Multiply $4$ by $- 2$ .

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} - 2 (- \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} - 2 (- \frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.3

Multiply $- 2 (- \frac{3}{5})$ .

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Step 2.2.2.1.1.3.1

Multiply $- 1$ by $- 2$ .

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} + 2 (\frac{3}{5})$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.3.2

Combine $2$ and $\frac{3}{5}$ .

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} + \frac{2 \cdot 3}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.3.3

Multiply $2$ by $3$ .

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y + \frac{- 8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.1.4

Move the negative in front of the fraction.

$\frac{4 (2 y - 9)}{5} = 2 y - \frac{8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = 2 y - \frac{8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 (2 y - 9)}{5} = 2 y \cdot \frac{5}{5} - \frac{8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.3

Combine $2 y$ and $\frac{5}{5}$ .

$\frac{4 (2 y - 9)}{5} = \frac{2 y \cdot 5}{5} - \frac{8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.4

Combine the numerators over the common denominator.

$\frac{4 (2 y - 9)}{5} = \frac{2 y \cdot 5 - 8 y}{5} + \frac{6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.5

Combine the numerators over the common denominator.

$\frac{4 (2 y - 9)}{5} = \frac{2 y \cdot 5 - 8 y + 6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.6

Multiply $5$ by $2$ .

$\frac{4 (2 y - 9)}{5} = \frac{10 y - 8 y + 6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.7

Subtract $8 y$ from $10 y$ .

$\frac{4 (2 y - 9)}{5} = \frac{2 y + 6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.8

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

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Step 2.2.2.1.8.1

Factor $2$ out of $2 y$ .

$\frac{4 (2 y - 9)}{5} = \frac{2 (y) + 6}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.8.2

Factor $2$ out of $6$ .

$\frac{4 (2 y - 9)}{5} = \frac{2 y + 2 \cdot 3}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 2.2.2.1.8.3

Factor $2$ out of $2 y + 2 \cdot 3$ .

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3

Solve for $y$ in $\frac{4 (2 y - 9)}{5} = \frac{2 (y + 3)}{5}$ .

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Step 3.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 (2 y - 9) = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.2

Simplify $4 (2 y - 9)$ .

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Step 3.2.1

Rewrite.

$0 + 0 + 4 (2 y - 9) = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.2.2

Simplify by adding zeros.

$4 (2 y - 9) = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.2.3

Apply the distributive property.

$4 (2 y) + 4 \cdot - 9 = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.2.4

Multiply.

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Step 3.2.4.1

Multiply $2$ by $4$ .

$8 y + 4 \cdot - 9 = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.2.4.2

Multiply $4$ by $- 9$ .

$8 y - 36 = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

$8 y - 36 = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

$8 y - 36 = 2 (y + 3)$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.3

Simplify $2 (y + 3)$ .

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Step 3.3.1

Apply the distributive property.

$8 y - 36 = 2 y + 2 \cdot 3$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.3.2

Multiply $2$ by $3$ .

$8 y - 36 = 2 y + 6$

$x = \frac{4 y}{5} - \frac{3}{5}$

$8 y - 36 = 2 y + 6$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.4

Move all terms containing $y$ to the left side of the equation.

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Step 3.4.1

Subtract $2 y$ from both sides of the equation.

$8 y - 36 - 2 y = 6$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.4.2

Subtract $2 y$ from $8 y$ .

$6 y - 36 = 6$

$x = \frac{4 y}{5} - \frac{3}{5}$

$6 y - 36 = 6$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.5

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

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Step 3.5.1

Add $36$ to both sides of the equation.

$6 y = 6 + 36$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.5.2

Add $6$ and $36$ .

$6 y = 42$

$x = \frac{4 y}{5} - \frac{3}{5}$

$6 y = 42$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.6

Divide each term in $6 y = 42$ by $6$ and simplify.

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Step 3.6.1

Divide each term in $6 y = 42$ by $6$ .

$\frac{6 y}{6} = \frac{42}{6}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $6$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{42}{6}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{42}{6}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$y = \frac{42}{6}$

$x = \frac{4 y}{5} - \frac{3}{5}$

$y = \frac{42}{6}$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Divide $42$ by $6$ .

$y = 7$

$x = \frac{4 y}{5} - \frac{3}{5}$

$y = 7$

$x = \frac{4 y}{5} - \frac{3}{5}$

$y = 7$

$x = \frac{4 y}{5} - \frac{3}{5}$

$y = 7$

$x = \frac{4 y}{5} - \frac{3}{5}$

Step 4

Replace all occurrences of $y$ with $7$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{4 y}{5} - \frac{3}{5}$ with $7$ .

$x = \frac{4 (7)}{5} - \frac{3}{5}$

$y = 7$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{4 (7)}{5} - \frac{3}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{4 (7) - 3}{5}$

$y = 7$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $4$ by $7$ .

$x = \frac{28 - 3}{5}$

$y = 7$

Step 4.2.1.2.2

Subtract $3$ from $28$ .

$x = \frac{25}{5}$

$y = 7$

Step 4.2.1.2.3

Divide $25$ by $5$ .

$x = 5$

$y = 7$

$x = 5$

$y = 7$

$x = 5$

$y = 7$

$x = 5$

$y = 7$

$x = 5$

$y = 7$

Step 5

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

$(5, 7)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(5, 7)$

Equation Form:

$x = 5, y = 7$

Step 7