Finite Math Examples

$3 x + 4 y = 9$ , $2 x - y = - 7$

Step 1

Solve for $x$ in $3 x + 4 y = 9$ .

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Step 1.1

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

$3 x = 9 - 4 y$

$2 x - y = - 7$

Step 1.2

Divide each term in $3 x = 9 - 4 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 9 - 4 y$ by $3$ .

$\frac{3 x}{3} = \frac{9}{3} + \frac{- 4 y}{3}$

$2 x - y = - 7$

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.

$\frac{3 x}{3} = \frac{9}{3} + \frac{- 4 y}{3}$

$2 x - y = - 7$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{3} + \frac{- 4 y}{3}$

$2 x - y = - 7$

$x = \frac{9}{3} + \frac{- 4 y}{3}$

$2 x - y = - 7$

$x = \frac{9}{3} + \frac{- 4 y}{3}$

$2 x - y = - 7$

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 $9$ by $3$ .

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

$2 x - y = - 7$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$2 x - y = - 7$

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

$2 x - y = - 7$

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

$2 x - y = - 7$

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

$2 x - y = - 7$

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

$2 x - y = - 7$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x - y = - 7$ with $3 - \frac{4 y}{3}$ .

$2 (3 - \frac{4 y}{3}) - y = - 7$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$2 \cdot 3 + 2 (- \frac{4 y}{3}) - y = - 7$

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

Step 2.2.1.1.2

Multiply $2$ by $3$ .

$6 + 2 (- \frac{4 y}{3}) - y = - 7$

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

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Multiply $- 1$ by $2$ .

$6 - 2 \frac{4 y}{3} - y = - 7$

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

Step 2.2.1.1.3.2

Combine $- 2$ and $\frac{4 y}{3}$ .

$6 + \frac{- 2 (4 y)}{3} - y = - 7$

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

Step 2.2.1.1.3.3

Multiply $4$ by $- 2$ .

$6 + \frac{- 8 y}{3} - y = - 7$

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

$6 + \frac{- 8 y}{3} - y = - 7$

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

$6 - \frac{8 y}{3} - y = - 7$

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

$6 - \frac{8 y}{3} - y = - 7$

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

Step 2.2.1.2

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

$6 - \frac{8 y}{3} - y \cdot \frac{3}{3} = - 7$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$6 - \frac{8 y}{3} + \frac{- y \cdot 3}{3} = - 7$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$6 + \frac{- 8 y - y \cdot 3}{3} = - 7$

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

$6 + \frac{- 8 y - y \cdot 3}{3} = - 7$

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

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 $y$ out of $- 8 y - y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $- 8 y$ .

$6 + \frac{y \cdot - 8 - y \cdot 3}{3} = - 7$

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

Step 2.2.1.4.1.1.2

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

$6 + \frac{y \cdot - 8 + y (- 1 \cdot 3)}{3} = - 7$

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

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot - 8 + y (- 1 \cdot 3)$ .

$6 + \frac{y (- 8 - 1 \cdot 3)}{3} = - 7$

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

$6 + \frac{y (- 8 - 1 \cdot 3)}{3} = - 7$

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

Step 2.2.1.4.1.2

Multiply $- 1$ by $3$ .

$6 + \frac{y (- 8 - 3)}{3} = - 7$

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

Step 2.2.1.4.1.3

Subtract $3$ from $- 8$ .

$6 + \frac{y \cdot - 11}{3} = - 7$

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

$6 + \frac{y \cdot - 11}{3} = - 7$

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

Step 2.2.1.4.2

Move $- 11$ to the left of $y$ .

$6 + \frac{- 11 \cdot y}{3} = - 7$

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

Step 2.2.1.4.3

Move the negative in front of the fraction.

$6 - \frac{11 y}{3} = - 7$

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

$6 - \frac{11 y}{3} = - 7$

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

$6 - \frac{11 y}{3} = - 7$

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

$6 - \frac{11 y}{3} = - 7$

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

$6 - \frac{11 y}{3} = - 7$

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

Step 3

Solve for $y$ in $6 - \frac{11 y}{3} = - 7$ .

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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 $6$ from both sides of the equation.

$- \frac{11 y}{3} = - 7 - 6$

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

Step 3.1.2

Subtract $6$ from $- 7$ .

$- \frac{11 y}{3} = - 13$

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

$- \frac{11 y}{3} = - 13$

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

Step 3.2

Multiply both sides of the equation by $- \frac{3}{11}$ .

$- \frac{3}{11} \cdot (- \frac{11 y}{3}) = - \frac{3}{11} \cdot - 13$

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

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 $- \frac{3}{11} (- \frac{11 y}{3})$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{3}{11}$ into the numerator.

$\frac{- 3}{11} \cdot (- \frac{11 y}{3}) = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{11 y}{3}$ into the numerator.

$\frac{- 3}{11} \cdot \frac{- 11 y}{3} = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{11} \cdot \frac{- 11 y}{3} = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{11} \cdot \frac{- 11 y}{3} = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{11} \cdot (- 11 y) = - \frac{3}{11} \cdot - 13$

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

$\frac{- 1}{11} \cdot (- 11 y) = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.2

Cancel the common factor of $11$ .

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Step 3.3.1.1.2.1

Factor $11$ out of $- 11 y$ .

$\frac{- 1}{11} \cdot (11 (- y)) = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{11} \cdot (11 (- y)) = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{3}{11} \cdot - 13$

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

$y = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{11} \cdot - 13$

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

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{3}{11} \cdot - 13$

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

$y = - \frac{3}{11} \cdot - 13$

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

$y = - \frac{3}{11} \cdot - 13$

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

$y = - \frac{3}{11} \cdot - 13$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- \frac{3}{11} \cdot - 13$ .

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Step 3.3.2.1.1

Multiply $- 13$ by $- 1$ .

$y = 13 (\frac{3}{11})$

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

Step 3.3.2.1.2

Combine $13$ and $\frac{3}{11}$ .

$y = \frac{13 \cdot 3}{11}$

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

Step 3.3.2.1.3

Multiply $13$ by $3$ .

$y = \frac{39}{11}$

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

$y = \frac{39}{11}$

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

$y = \frac{39}{11}$

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

$y = \frac{39}{11}$

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

$y = \frac{39}{11}$

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

Step 4

Replace all occurrences of $y$ with $\frac{39}{11}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 3 - \frac{4 y}{3}$ with $\frac{39}{11}$ .

$x = 3 - \frac{4 (\frac{39}{11})}{3}$

$y = \frac{39}{11}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $3 - \frac{4 (\frac{39}{11})}{3}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Combine $4$ and $\frac{39}{11}$ .

$x = 3 - \frac{\frac{4 \cdot 39}{11}}{3}$

$y = \frac{39}{11}$

Step 4.2.1.1.2

Multiply $4$ by $39$ .

$x = 3 - \frac{\frac{156}{11}}{3}$

$y = \frac{39}{11}$

Step 4.2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = 3 - (\frac{156}{11} \cdot \frac{1}{3})$

$y = \frac{39}{11}$

Step 4.2.1.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.1.4.1

Factor $3$ out of $156$ .

$x = 3 - (\frac{3 (52)}{11} \cdot \frac{1}{3})$

$y = \frac{39}{11}$

Step 4.2.1.1.4.2

Cancel the common factor.

$x = 3 - (\frac{3 \cdot 52}{11} \cdot \frac{1}{3})$

$y = \frac{39}{11}$

Step 4.2.1.1.4.3

Rewrite the expression.

$x = 3 - \frac{52}{11}$

$y = \frac{39}{11}$

$x = 3 - \frac{52}{11}$

$y = \frac{39}{11}$

$x = 3 - \frac{52}{11}$

$y = \frac{39}{11}$

Step 4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$x = 3 \cdot \frac{11}{11} - \frac{52}{11}$

$y = \frac{39}{11}$

Step 4.2.1.3

Combine $3$ and $\frac{11}{11}$ .

$x = \frac{3 \cdot 11}{11} - \frac{52}{11}$

$y = \frac{39}{11}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 11 - 52}{11}$

$y = \frac{39}{11}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $3$ by $11$ .

$x = \frac{33 - 52}{11}$

$y = \frac{39}{11}$

Step 4.2.1.5.2

Subtract $52$ from $33$ .

$x = \frac{- 19}{11}$

$y = \frac{39}{11}$

$x = \frac{- 19}{11}$

$y = \frac{39}{11}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x = - \frac{19}{11}$

$y = \frac{39}{11}$

$x = - \frac{19}{11}$

$y = \frac{39}{11}$

$x = - \frac{19}{11}$

$y = \frac{39}{11}$

$x = - \frac{19}{11}$

$y = \frac{39}{11}$

Step 5

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

$(- \frac{19}{11}, \frac{39}{11})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{19}{11}, \frac{39}{11})$

Equation Form:

$x = - \frac{19}{11}, y = \frac{39}{11}$

Step 7