Finite Math Examples

$- 8 x + 4 y = 36$ , $5 x - 3 y = - 23$

Step 1

Solve for $x$ in $- 8 x + 4 y = 36$ .

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Step 1.1

Subtract $4 y$ from both sides of the equation.

$- 8 x = 36 - 4 y$

$5 x - 3 y = - 23$

Step 1.2

Divide each term in $- 8 x = 36 - 4 y$ by $- 8$ and simplify.

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Step 1.2.1

Divide each term in $- 8 x = 36 - 4 y$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{36}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{36}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

$x = \frac{36}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

$x = \frac{36}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

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 $36$ and $- 8$ .

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Step 1.2.3.1.1.1

Factor $4$ out of $36$ .

$x = \frac{4 (9)}{- 8} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $4$ out of $- 8$ .

$x = \frac{4 \cdot 9}{4 \cdot - 2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 9}{4 \cdot - 2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{9}{- 2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

$x = \frac{9}{- 2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

$x = \frac{9}{- 2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{9}{2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.3

Cancel the common factor of $- 4$ and $- 8$ .

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Step 1.2.3.1.3.1

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

$x = - \frac{9}{2} + \frac{- 4 y}{- 8}$

$5 x - 3 y = - 23$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

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

$x = - \frac{9}{2} + \frac{- 4 y}{- 4 \cdot 2}$

$5 x - 3 y = - 23$

Step 1.2.3.1.3.2.2

Cancel the common factor.

$x = - \frac{9}{2} + \frac{- 4 y}{- 4 \cdot 2}$

$5 x - 3 y = - 23$

Step 1.2.3.1.3.2.3

Rewrite the expression.

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

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

$5 x - 3 y = - 23$

Step 2

Replace all occurrences of $x$ with $- \frac{9}{2} + \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x - 3 y = - 23$ with $- \frac{9}{2} + \frac{y}{2}$ .

$5 (- \frac{9}{2} + \frac{y}{2}) - 3 y = - 23$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (- \frac{9}{2} + \frac{y}{2}) - 3 y$ .

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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.

$5 (- \frac{9}{2}) + 5 (\frac{y}{2}) - 3 y = - 23$

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

Step 2.2.1.1.2

Multiply $5 (- \frac{9}{2})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{9}{2}) + 5 (\frac{y}{2}) - 3 y = - 23$

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

Step 2.2.1.1.2.2

Combine $- 5$ and $\frac{9}{2}$ .

$\frac{- 5 \cdot 9}{2} + 5 (\frac{y}{2}) - 3 y = - 23$

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

Step 2.2.1.1.2.3

Multiply $- 5$ by $9$ .

$\frac{- 45}{2} + 5 (\frac{y}{2}) - 3 y = - 23$

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

$\frac{- 45}{2} + 5 (\frac{y}{2}) - 3 y = - 23$

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

Step 2.2.1.1.3

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

$\frac{- 45}{2} + \frac{5 y}{2} - 3 y = - 23$

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{45}{2} + \frac{5 y}{2} - 3 y = - 23$

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

$- \frac{45}{2} + \frac{5 y}{2} - 3 y = - 23$

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

Step 2.2.1.2

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

$- \frac{45}{2} + \frac{5 y}{2} - 3 y \cdot \frac{2}{2} = - 23$

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

Step 2.2.1.3

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

$- \frac{45}{2} + \frac{5 y}{2} + \frac{- 3 y \cdot 2}{2} = - 23$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{45}{2} + \frac{5 y - 3 y \cdot 2}{2} = - 23$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 45 + 5 y - 3 y \cdot 2}{2} = - 23$

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

Step 2.2.1.6

Multiply $2$ by $- 3$ .

$\frac{- 45 + 5 y - 6 y}{2} = - 23$

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

Step 2.2.1.7

Subtract $6 y$ from $5 y$ .

$\frac{- 45 - y}{2} = - 23$

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

Step 2.2.1.8

Rewrite $- 45$ as $- 1 (45)$ .

$\frac{- 1 \cdot 45 - y}{2} = - 23$

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

Step 2.2.1.9

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

$\frac{- 1 \cdot 45 - (y)}{2} = - 23$

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

Step 2.2.1.10

Factor $- 1$ out of $- 1 (45) - (y)$ .

$\frac{- 1 (45 + y)}{2} = - 23$

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

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{45 + y}{2} = - 23$

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

$- \frac{45 + y}{2} = - 23$

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

$- \frac{45 + y}{2} = - 23$

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

$- \frac{45 + y}{2} = - 23$

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

Step 3

Solve for $y$ in $- \frac{45 + y}{2} = - 23$ .

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Step 3.1

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{45 + y}{2}) = - 2 \cdot - 23$

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

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 $- 2 (- \frac{45 + y}{2})$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{45 + y}{2}$ into the numerator.

$- 2 \frac{- (45 + y)}{2} = - 2 \cdot - 23$

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

Step 3.2.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- (45 + y)}{2}) = - 2 \cdot - 23$

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

Step 3.2.1.1.1.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- (45 + y)}{2}) = - 2 \cdot - 23$

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

Step 3.2.1.1.1.4

Rewrite the expression.

$45 + y = - 2 \cdot - 23$

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

$45 + y = - 2 \cdot - 23$

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

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (45 + y) = - 2 \cdot - 23$

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

Step 3.2.1.1.2.2

Multiply $45 + y$ by $1$ .

$45 + y = - 2 \cdot - 23$

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

$45 + y = - 2 \cdot - 23$

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

$45 + y = - 2 \cdot - 23$

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

$45 + y = - 2 \cdot - 23$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 2$ by $- 23$ .

$45 + y = 46$

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

$45 + y = 46$

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

$45 + y = 46$

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

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 $45$ from both sides of the equation.

$y = 46 - 45$

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

Step 3.3.2

Subtract $45$ from $46$ .

$y = 1$

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

$y = 1$

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

$y = 1$

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

Step 4

Replace all occurrences of $y$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{9}{2} + \frac{y}{2}$ with $1$ .

$x = - \frac{9}{2} + \frac{1}{2}$

$y = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{9}{2} + \frac{1}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 9 + 1}{2}$

$y = 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Add $- 9$ and $1$ .

$x = \frac{- 8}{2}$

$y = 1$

Step 4.2.1.2.2

Divide $- 8$ by $2$ .

$x = - 4$

$y = 1$

$x = - 4$

$y = 1$

$x = - 4$

$y = 1$

$x = - 4$

$y = 1$

$x = - 4$

$y = 1$

Step 5

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

$(- 4, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 4, 1)$

Equation Form:

$x = - 4, y = 1$

Step 7