Finite Math Examples

$12 x - 5 y = 9$ , $3 x - 8 y = - 18$

Step 1

Solve for $x$ in $12 x - 5 y = 9$ .

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Step 1.1

Add $5 y$ to both sides of the equation.

$12 x = 9 + 5 y$

$3 x - 8 y = - 18$

Step 1.2

Divide each term in $12 x = 9 + 5 y$ by $12$ and simplify.

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Step 1.2.1

Divide each term in $12 x = 9 + 5 y$ by $12$ .

$\frac{12 x}{12} = \frac{9}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $12$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{9}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{9}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{9}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $9$ and $12$ .

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Step 1.2.3.1.1

Factor $3$ out of $9$ .

$x = \frac{3 (3)}{12} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Factor $3$ out of $12$ .

$x = \frac{3 \cdot 3}{3 \cdot 4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 3}{3 \cdot 4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 1.2.3.1.2.3

Rewrite the expression.

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$3 x - 8 y = - 18$

Step 2

Replace all occurrences of $x$ with $\frac{3}{4} + \frac{5 y}{12}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 8 y = - 18$ with $\frac{3}{4} + \frac{5 y}{12}$ .

$3 (\frac{3}{4} + \frac{5 y}{12}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$3 (\frac{3}{4}) + 3 (\frac{5 y}{12}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.1.2

Multiply $3 (\frac{3}{4})$ .

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Step 2.2.1.1.2.1

Combine $3$ and $\frac{3}{4}$ .

$\frac{3 \cdot 3}{4} + 3 (\frac{5 y}{12}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.1.2.2

Multiply $3$ by $3$ .

$\frac{9}{4} + 3 (\frac{5 y}{12}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9}{4} + 3 (\frac{5 y}{12}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Factor $3$ out of $12$ .

$\frac{9}{4} + 3 (\frac{5 y}{3 (4)}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.1.3.2

Cancel the common factor.

$\frac{9}{4} + 3 (\frac{5 y}{3 \cdot 4}) - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.1.3.3

Rewrite the expression.

$\frac{9}{4} + \frac{5 y}{4} - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9}{4} + \frac{5 y}{4} - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9}{4} + \frac{5 y}{4} - 8 y = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.2

To write $- 8 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{9}{4} + \frac{5 y}{4} - 8 y \cdot \frac{4}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.3

Combine $- 8 y$ and $\frac{4}{4}$ .

$\frac{9}{4} + \frac{5 y}{4} + \frac{- 8 y \cdot 4}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{9}{4} + \frac{5 y - 8 y \cdot 4}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{9 + 5 y - 8 y \cdot 4}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.6

Multiply $4$ by $- 8$ .

$\frac{9 + 5 y - 32 y}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.7

Subtract $32 y$ from $5 y$ .

$\frac{9 - 27 y}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.8

Factor $9$ out of $9 - 27 y$ .

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Step 2.2.1.8.1

Factor $9$ out of $9$ .

$\frac{9 (1) - 27 y}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.8.2

Factor $9$ out of $- 27 y$ .

$\frac{9 (1) + 9 (- 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 2.2.1.8.3

Factor $9$ out of $9 (1) + 9 (- 3 y)$ .

$\frac{9 (1 - 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9 (1 - 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9 (1 - 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9 (1 - 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

$\frac{9 (1 - 3 y)}{4} = - 18$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3

Solve for $y$ in $\frac{9 (1 - 3 y)}{4} = - 18$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{9 (1 - 3 y)}{4} \cdot 4 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

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 $\frac{9 (1 - 3 y)}{4} \cdot 4$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{9 (1 - 3 y)}{4} \cdot 4 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.1.1.1.2

Rewrite the expression.

$9 (1 - 3 y) = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

$9 (1 - 3 y) = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.1.1.2

Apply the distributive property.

$9 \cdot 1 + 9 (- 3 y) = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.1.1.3

Simplify the expression.

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Step 3.2.1.1.3.1

Multiply $9$ by $1$ .

$9 + 9 (- 3 y) = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.1.1.3.2

Multiply $- 3$ by $9$ .

$9 - 27 y = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.1.1.3.3

Reorder $9$ and $- 27 y$ .

$- 27 y + 9 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y + 9 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y + 9 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y + 9 = - 18 \cdot 4$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 18$ by $4$ .

$- 27 y + 9 = - 72$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y + 9 = - 72$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y + 9 = - 72$

$x = \frac{3}{4} + \frac{5 y}{12}$

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 $9$ from both sides of the equation.

$- 27 y = - 72 - 9$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.3.1.2

Subtract $9$ from $- 72$ .

$- 27 y = - 81$

$x = \frac{3}{4} + \frac{5 y}{12}$

$- 27 y = - 81$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.3.2

Divide each term in $- 27 y = - 81$ by $- 27$ and simplify.

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Step 3.3.2.1

Divide each term in $- 27 y = - 81$ by $- 27$ .

$\frac{- 27 y}{- 27} = \frac{- 81}{- 27}$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 27$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 27 y}{- 27} = \frac{- 81}{- 27}$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 81}{- 27}$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = \frac{- 81}{- 27}$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = \frac{- 81}{- 27}$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $- 81$ by $- 27$ .

$y = 3$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = 3$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = 3$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = 3$

$x = \frac{3}{4} + \frac{5 y}{12}$

$y = 3$

$x = \frac{3}{4} + \frac{5 y}{12}$

Step 4

Replace all occurrences of $y$ with $3$ in each equation.

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Step 4.1

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

$x = \frac{3}{4} + \frac{5 (3)}{12}$

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{4} + \frac{5 (3)}{12}$ .

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Step 4.2.1.1

Multiply $5$ by $3$ .

$x = \frac{3}{4} + \frac{15}{12}$

$y = 3$

Step 4.2.1.2

Cancel the common factor of $15$ and $12$ .

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Step 4.2.1.2.1

Factor $3$ out of $15$ .

$x = \frac{3}{4} + \frac{3 (5)}{12}$

$y = 3$

Step 4.2.1.2.2

Cancel the common factors.

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Step 4.2.1.2.2.1

Factor $3$ out of $12$ .

$x = \frac{3}{4} + \frac{3 \cdot 5}{3 \cdot 4}$

$y = 3$

Step 4.2.1.2.2.2

Cancel the common factor.

$x = \frac{3}{4} + \frac{3 \cdot 5}{3 \cdot 4}$

$y = 3$

Step 4.2.1.2.2.3

Rewrite the expression.

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

$y = 3$

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

$y = 3$

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

$y = 3$

Step 4.2.1.3

Combine the numerators over the common denominator.

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

$y = 3$

Step 4.2.1.4

Simplify the expression.

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Step 4.2.1.4.1

Add $3$ and $5$ .

$x = \frac{8}{4}$

$y = 3$

Step 4.2.1.4.2

Divide $8$ by $4$ .

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

$x = 2$

$y = 3$

Step 5

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

$(2, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 3)$

Equation Form:

$x = 2, y = 3$

Step 7