Finite Math Examples

$4 x + 2 y = 20$ , $9 x + 9 y = 36$

Step 1

Solve for $x$ in $4 x + 2 y = 20$ .

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Step 1.1

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

$4 x = 20 - 2 y$

$9 x + 9 y = 36$

Step 1.2

Divide each term in $4 x = 20 - 2 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 20 - 2 y$ by $4$ .

$\frac{4 x}{4} = \frac{20}{4} + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{20}{4} + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{4} + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

$x = \frac{20}{4} + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

$x = \frac{20}{4} + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

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

Divide $20$ by $4$ .

$x = 5 + \frac{- 2 y}{4}$

$9 x + 9 y = 36$

Step 1.2.3.1.2

Cancel the common factor of $- 2$ and $4$ .

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Step 1.2.3.1.2.1

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

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

$9 x + 9 y = 36$

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 $4$ .

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

$9 x + 9 y = 36$

Step 1.2.3.1.2.2.2

Cancel the common factor.

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

$9 x + 9 y = 36$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 5 + \frac{- y}{2}$

$9 x + 9 y = 36$

$x = 5 + \frac{- y}{2}$

$9 x + 9 y = 36$

$x = 5 + \frac{- y}{2}$

$9 x + 9 y = 36$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = 5 - \frac{y}{2}$

$9 x + 9 y = 36$

$x = 5 - \frac{y}{2}$

$9 x + 9 y = 36$

$x = 5 - \frac{y}{2}$

$9 x + 9 y = 36$

$x = 5 - \frac{y}{2}$

$9 x + 9 y = 36$

$x = 5 - \frac{y}{2}$

$9 x + 9 y = 36$

Step 2

Replace all occurrences of $x$ with $5 - \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $9 x + 9 y = 36$ with $5 - \frac{y}{2}$ .

$9 (5 - \frac{y}{2}) + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $9 (5 - \frac{y}{2}) + 9 y$ .

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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.

$9 \cdot 5 + 9 (- \frac{y}{2}) + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.1.2

Multiply $9$ by $5$ .

$45 + 9 (- \frac{y}{2}) + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.1.3

Multiply $9 (- \frac{y}{2})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $9$ .

$45 - 9 \frac{y}{2} + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.1.3.2

Combine $- 9$ and $\frac{y}{2}$ .

$45 + \frac{- 9 y}{2} + 9 y = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{- 9 y}{2} + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$45 - \frac{9 y}{2} + 9 y = 36$

$x = 5 - \frac{y}{2}$

$45 - \frac{9 y}{2} + 9 y = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.2

To write $9 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$45 - \frac{9 y}{2} + 9 y \cdot \frac{2}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $9 y$ and $\frac{2}{2}$ .

$45 - \frac{9 y}{2} + \frac{9 y \cdot 2}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$45 + \frac{- 9 y + 9 y \cdot 2}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{- 9 y + 9 y \cdot 2}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Factor $9 y$ out of $- 9 y + 9 y \cdot 2$ .

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Step 2.2.1.4.1.1

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

$45 + \frac{9 y (- 1) + 9 y \cdot 2}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.4.1.2

Factor $9 y$ out of $9 y \cdot 2$ .

$45 + \frac{9 y (- 1) + 9 y (2)}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.4.1.3

Factor $9 y$ out of $9 y (- 1) + 9 y (2)$ .

$45 + \frac{9 y (- 1 + 2)}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{9 y (- 1 + 2)}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.4.2

Add $- 1$ and $2$ .

$45 + \frac{9 y \cdot 1}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 2.2.1.4.3

Multiply $9$ by $1$ .

$45 + \frac{9 y}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{9 y}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{9 y}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{9 y}{2} = 36$

$x = 5 - \frac{y}{2}$

$45 + \frac{9 y}{2} = 36$

$x = 5 - \frac{y}{2}$

Step 3

Solve for $y$ in $45 + \frac{9 y}{2} = 36$ .

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Step 3.1

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

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Step 3.1.1

Subtract $45$ from both sides of the equation.

$\frac{9 y}{2} = 36 - 45$

$x = 5 - \frac{y}{2}$

Step 3.1.2

Subtract $45$ from $36$ .

$\frac{9 y}{2} = - 9$

$x = 5 - \frac{y}{2}$

$\frac{9 y}{2} = - 9$

$x = 5 - \frac{y}{2}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{9}$ .

$\frac{2}{9} \cdot \frac{9 y}{2} = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{2}{9} \cdot \frac{9 y}{2}$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{2}{9} \cdot \frac{9 y}{2} = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{9} \cdot (9 y) = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

$\frac{1}{9} \cdot (9 y) = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3.1.1.2

Cancel the common factor of $9$ .

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Step 3.3.1.1.2.1

Factor $9$ out of $9 y$ .

$\frac{1}{9} \cdot (9 (y)) = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{9} \cdot (9 y) = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

$y = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

$y = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

$y = \frac{2}{9} \cdot - 9$

$x = 5 - \frac{y}{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{2}{9} \cdot - 9$ .

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Step 3.3.2.1.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1.1

Factor $9$ out of $- 9$ .

$y = \frac{2}{9} \cdot (9 (- 1))$

$x = 5 - \frac{y}{2}$

Step 3.3.2.1.1.2

Cancel the common factor.

$y = \frac{2}{9} \cdot (9 \cdot - 1)$

$x = 5 - \frac{y}{2}$

Step 3.3.2.1.1.3

Rewrite the expression.

$y = 2 \cdot - 1$

$x = 5 - \frac{y}{2}$

$y = 2 \cdot - 1$

$x = 5 - \frac{y}{2}$

Step 3.3.2.1.2

Multiply $2$ by $- 1$ .

$y = - 2$

$x = 5 - \frac{y}{2}$

$y = - 2$

$x = 5 - \frac{y}{2}$

$y = - 2$

$x = 5 - \frac{y}{2}$

$y = - 2$

$x = 5 - \frac{y}{2}$

$y = - 2$

$x = 5 - \frac{y}{2}$

Step 4

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 4.1

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

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

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $5 - \frac{- 2}{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Divide $- 2$ by $2$ .

$x = 5 + 1$

$y = - 2$

Step 4.2.1.1.2

Multiply $- 1$ by $- 1$ .

$x = 5 + 1$

$y = - 2$

$x = 5 + 1$

$y = - 2$

Step 4.2.1.2

Add $5$ and $1$ .

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

$x = 6$

$y = - 2$

Step 5

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

$(6, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, - 2)$

Equation Form:

$x = 6, y = - 2$

Step 7