Finite Math Examples

$3 x + 4 y = - 5$ , $9 x - 2 y = 13$

Step 1

Solve for $x$ in $3 x + 4 y = - 5$ .

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Step 1.1

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

$3 x = - 5 - 4 y$

$9 x - 2 y = 13$

Step 1.2

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

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Step 1.2.1

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

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

$9 x - 2 y = 13$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$9 x - 2 y = 13$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

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

Move the negative in front of the fraction.

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

$9 x - 2 y = 13$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

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

$9 x - 2 y = 13$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $9 x - 2 y = 13$ with $- \frac{5}{3} - \frac{4 y}{3}$ .

$9 (- \frac{5}{3} - \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$9 (- \frac{5}{3}) + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$9 (\frac{- 5}{3}) + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.2.2

Factor $3$ out of $9$ .

$3 (3) (\frac{- 5}{3}) + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.2.3

Cancel the common factor.

$3 \cdot (3 (\frac{- 5}{3})) + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.2.4

Rewrite the expression.

$3 \cdot - 5 + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

$3 \cdot - 5 + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.3

Multiply $3$ by $- 5$ .

$- 15 + 9 (- \frac{4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{4 y}{3}$ into the numerator.

$- 15 + 9 (\frac{- 4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.4.2

Factor $3$ out of $9$ .

$- 15 + 3 (3) (\frac{- 4 y}{3}) - 2 y = 13$

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

Step 2.2.1.1.4.3

Cancel the common factor.

$- 15 + 3 \cdot (3 (\frac{- 4 y}{3})) - 2 y = 13$

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

Step 2.2.1.1.4.4

Rewrite the expression.

$- 15 + 3 (- 4 y) - 2 y = 13$

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

$- 15 + 3 (- 4 y) - 2 y = 13$

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

Step 2.2.1.1.5

Multiply $- 4$ by $3$ .

$- 15 - 12 y - 2 y = 13$

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

$- 15 - 12 y - 2 y = 13$

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

Step 2.2.1.2

Subtract $2 y$ from $- 12 y$ .

$- 15 - 14 y = 13$

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

$- 15 - 14 y = 13$

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

$- 15 - 14 y = 13$

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

$- 15 - 14 y = 13$

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

Step 3

Solve for $y$ in $- 15 - 14 y = 13$ .

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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 $15$ to both sides of the equation.

$- 14 y = 13 + 15$

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

Step 3.1.2

Add $13$ and $15$ .

$- 14 y = 28$

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

$- 14 y = 28$

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

Step 3.2

Divide each term in $- 14 y = 28$ by $- 14$ and simplify.

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Step 3.2.1

Divide each term in $- 14 y = 28$ by $- 14$ .

$\frac{- 14 y}{- 14} = \frac{28}{- 14}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 14$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 14 y}{- 14} = \frac{28}{- 14}$

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{28}{- 14}$

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

$y = \frac{28}{- 14}$

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

$y = \frac{28}{- 14}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $28$ by $- 14$ .

$y = - 2$

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

$y = - 2$

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

$y = - 2$

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

$y = - 2$

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

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

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

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Combine the numerators over the common denominator.

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

$y = - 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 5 + 8}{3}$

$y = - 2$

Step 4.2.1.2.2

Add $- 5$ and $8$ .

$x = \frac{3}{3}$

$y = - 2$

Step 4.2.1.2.3

Divide $3$ by $3$ .

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

Step 5

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

$(1, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - 2)$

Equation Form:

$x = 1, y = - 2$

Step 7