Finite Math Examples

$4 x + 3 y = - 10$ , $5 x - 3 y = - 53$

Step 1

Solve for $x$ in $4 x + 3 y = - 10$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$4 x = - 10 - 3 y$

$5 x - 3 y = - 53$

Step 1.2

Divide each term in $4 x = - 10 - 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 10 - 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 10}{4} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

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.

$\frac{4 x}{4} = \frac{- 10}{4} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

$x = \frac{- 10}{4} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

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 $- 10$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 10$ .

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

$5 x - 3 y = - 53$

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 $4$ .

$x = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

$x = \frac{- 5}{2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{5}{2} + \frac{- 3 y}{4}$

$5 x - 3 y = - 53$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{5}{2} - \frac{3 y}{4}$

$5 x - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$5 x - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$5 x - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$5 x - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$5 x - 3 y = - 53$

Step 2

Replace all occurrences of $x$ with $- \frac{5}{2} - \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x - 3 y = - 53$ with $- \frac{5}{2} - \frac{3 y}{4}$ .

$5 (- \frac{5}{2} - \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$5 (- \frac{5}{2}) + 5 (- \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2

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

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Step 2.2.1.1.2.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{5}{2}) + 5 (- \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2.2

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

$\frac{- 5 \cdot 5}{2} + 5 (- \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.2.3

Multiply $- 5$ by $5$ .

$\frac{- 25}{2} + 5 (- \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$\frac{- 25}{2} + 5 (- \frac{3 y}{4}) - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Multiply $- 1$ by $5$ .

$\frac{- 25}{2} - 5 \frac{3 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3.2

Combine $- 5$ and $\frac{3 y}{4}$ .

$\frac{- 25}{2} + \frac{- 5 (3 y)}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.3.3

Multiply $3$ by $- 5$ .

$\frac{- 25}{2} + \frac{- 15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$\frac{- 25}{2} + \frac{- 15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{25}{2} + \frac{- 15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{25}{2} - \frac{15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{15 y}{4} - 3 y = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.2

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{25}{2} - \frac{15 y}{4} - 3 y \cdot \frac{4}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- \frac{25}{2} - \frac{15 y}{4} + \frac{- 3 y \cdot 4}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- \frac{25}{2} + \frac{- 15 y - 3 y \cdot 4}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} + \frac{- 15 y - 3 y \cdot 4}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $3 y$ out of $- 15 y - 3 y \cdot 4$ .

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Step 2.2.1.4.1.1.1

Factor $3 y$ out of $- 15 y$ .

$- \frac{25}{2} + \frac{3 y (- 5) - 3 y \cdot 4}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.1.2

Factor $3 y$ out of $- 3 y \cdot 4$ .

$- \frac{25}{2} + \frac{3 y (- 5) + 3 y (- 1 \cdot 4)}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.1.3

Factor $3 y$ out of $3 y (- 5) + 3 y (- 1 \cdot 4)$ .

$- \frac{25}{2} + \frac{3 y (- 5 - 1 \cdot 4)}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} + \frac{3 y (- 5 - 1 \cdot 4)}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.2

Multiply $- 1$ by $4$ .

$- \frac{25}{2} + \frac{3 y (- 5 - 4)}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.3

Subtract $4$ from $- 5$ .

$- \frac{25}{2} + \frac{3 y \cdot - 9}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.1.4

Multiply $- 9$ by $3$ .

$- \frac{25}{2} + \frac{- 27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} + \frac{- 27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$- \frac{25}{2} - \frac{27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{25}{2} - \frac{27 y}{4} = - 53$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3

Solve for $y$ in $- \frac{25}{2} - \frac{27 y}{4} = - 53$ .

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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

Add $\frac{25}{2}$ to both sides of the equation.

$- \frac{27 y}{4} = - 53 + \frac{25}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.2

To write $- 53$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{27 y}{4} = - 53 \cdot \frac{2}{2} + \frac{25}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.3

Combine $- 53$ and $\frac{2}{2}$ .

$- \frac{27 y}{4} = \frac{- 53 \cdot 2}{2} + \frac{25}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.4

Combine the numerators over the common denominator.

$- \frac{27 y}{4} = \frac{- 53 \cdot 2 + 25}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $- 53$ by $2$ .

$- \frac{27 y}{4} = \frac{- 106 + 25}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.5.2

Add $- 106$ and $25$ .

$- \frac{27 y}{4} = \frac{- 81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{27 y}{4} = \frac{- 81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.1.6

Move the negative in front of the fraction.

$- \frac{27 y}{4} = - \frac{81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$- \frac{27 y}{4} = - \frac{81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.2

Multiply both sides of the equation by $- \frac{4}{27}$ .

$- \frac{4}{27} \cdot (- \frac{27 y}{4}) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

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 $- \frac{4}{27} (- \frac{27 y}{4})$ .

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{4}{27}$ into the numerator.

$\frac{- 4}{27} \cdot (- \frac{27 y}{4}) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{27 y}{4}$ into the numerator.

$\frac{- 4}{27} \cdot \frac{- 27 y}{4} = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{27} \cdot \frac{- 27 y}{4} = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{27} \cdot \frac{- 27 y}{4} = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{27} \cdot (- 27 y) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$\frac{- 1}{27} \cdot (- 27 y) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.2

Cancel the common factor of $27$ .

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Step 3.3.1.1.2.1

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

$\frac{- 1}{27} \cdot (27 (- y)) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{27} \cdot (27 (- y)) = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - \frac{4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{4}{27} (- \frac{81}{2})$ .

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{4}{27}$ into the numerator.

$y = \frac{- 4}{27} \cdot (- \frac{81}{2})$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.2

Move the leading negative in $- \frac{81}{2}$ into the numerator.

$y = \frac{- 4}{27} \cdot \frac{- 81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.3

Factor $2$ out of $- 4$ .

$y = \frac{2 (- 2)}{27} \cdot \frac{- 81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.4

Cancel the common factor.

$y = \frac{2 \cdot - 2}{27} \cdot \frac{- 81}{2}$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.1.5

Rewrite the expression.

$y = \frac{- 2}{27} \cdot - 81$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = \frac{- 2}{27} \cdot - 81$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.2

Cancel the common factor of $27$ .

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Step 3.3.2.1.2.1

Factor $27$ out of $- 81$ .

$y = \frac{- 2}{27} \cdot (27 (- 3))$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.2.2

Cancel the common factor.

$y = \frac{- 2}{27} \cdot (27 \cdot - 3)$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.2.3

Rewrite the expression.

$y = - 2 \cdot - 3$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = - 2 \cdot - 3$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 3.3.2.1.3

Multiply $- 2$ by $- 3$ .

$y = 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

$y = 6$

$x = - \frac{5}{2} - \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{5}{2} - \frac{3 y}{4}$ with $6$ .

$x = - \frac{5}{2} - \frac{3 (6)}{4}$

$y = 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{5}{2} - \frac{3 (6)}{4}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $6$ and $4$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (6)$ .

$x = - \frac{5}{2} - \frac{2 (3 \cdot (3))}{4}$

$y = 6$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $4$ .

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

$y = 6$

Step 4.2.1.1.1.2.2

Cancel the common factor.

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

$y = 6$

Step 4.2.1.1.1.2.3

Rewrite the expression.

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

$y = 6$

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

$y = 6$

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

$y = 6$

Step 4.2.1.1.2

Multiply $3$ by $3$ .

$x = - \frac{5}{2} - \frac{9}{2}$

$y = 6$

$x = - \frac{5}{2} - \frac{9}{2}$

$y = 6$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{- 5 - 9}{2}$

$y = 6$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Subtract $9$ from $- 5$ .

$x = \frac{- 14}{2}$

$y = 6$

Step 4.2.1.2.2.2

Divide $- 14$ by $2$ .

$x = - 7$

$y = 6$

$x = - 7$

$y = 6$

$x = - 7$

$y = 6$

$x = - 7$

$y = 6$

$x = - 7$

$y = 6$

$x = - 7$

$y = 6$

Step 5

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

$(- 7, 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 7, 6)$

Equation Form:

$x = - 7, y = 6$

Step 7