Finite Math Examples

$3 x + 8 y = - 7$ , $5 x - 8 y = 8$

Step 1

Solve for $x$ in $3 x + 8 y = - 7$ .

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Step 1.1

Subtract $8 y$ from both sides of the equation.

$3 x = - 7 - 8 y$

$5 x - 8 y = 8$

Step 1.2

Divide each term in $3 x = - 7 - 8 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 7 - 8 y$ by $3$ .

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

$5 x - 8 y = 8$

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.

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

$5 x - 8 y = 8$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

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.

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

$5 x - 8 y = 8$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

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

$5 x - 8 y = 8$

Step 2

Replace all occurrences of $x$ with $- \frac{7}{3} - \frac{8 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x - 8 y = 8$ with $- \frac{7}{3} - \frac{8 y}{3}$ .

$5 (- \frac{7}{3} - \frac{8 y}{3}) - 8 y = 8$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (- \frac{7}{3} - \frac{8 y}{3}) - 8 y$ .

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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.

$5 (- \frac{7}{3}) + 5 (- \frac{8 y}{3}) - 8 y = 8$

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

Step 2.2.1.1.2

Multiply $5 (- \frac{7}{3})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{7}{3}) + 5 (- \frac{8 y}{3}) - 8 y = 8$

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

Step 2.2.1.1.2.2

Combine $- 5$ and $\frac{7}{3}$ .

$\frac{- 5 \cdot 7}{3} + 5 (- \frac{8 y}{3}) - 8 y = 8$

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

Step 2.2.1.1.2.3

Multiply $- 5$ by $7$ .

$\frac{- 35}{3} + 5 (- \frac{8 y}{3}) - 8 y = 8$

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

$\frac{- 35}{3} + 5 (- \frac{8 y}{3}) - 8 y = 8$

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

Step 2.2.1.1.3

Multiply $5 (- \frac{8 y}{3})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $5$ .

$\frac{- 35}{3} - 5 \frac{8 y}{3} - 8 y = 8$

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

Step 2.2.1.1.3.2

Combine $- 5$ and $\frac{8 y}{3}$ .

$\frac{- 35}{3} + \frac{- 5 (8 y)}{3} - 8 y = 8$

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

Step 2.2.1.1.3.3

Multiply $8$ by $- 5$ .

$\frac{- 35}{3} + \frac{- 40 y}{3} - 8 y = 8$

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

$\frac{- 35}{3} + \frac{- 40 y}{3} - 8 y = 8$

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

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.

$- \frac{35}{3} + \frac{- 40 y}{3} - 8 y = 8$

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

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{35}{3} - \frac{40 y}{3} - 8 y = 8$

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

$- \frac{35}{3} - \frac{40 y}{3} - 8 y = 8$

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

$- \frac{35}{3} - \frac{40 y}{3} - 8 y = 8$

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

Step 2.2.1.2

To write $- 8 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{35}{3} - \frac{40 y}{3} - 8 y \cdot \frac{3}{3} = 8$

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

Step 2.2.1.3

Combine $- 8 y$ and $\frac{3}{3}$ .

$- \frac{35}{3} - \frac{40 y}{3} + \frac{- 8 y \cdot 3}{3} = 8$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{35}{3} + \frac{- 40 y - 8 y \cdot 3}{3} = 8$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 35 - 40 y - 8 y \cdot 3}{3} = 8$

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

Step 2.2.1.6

Multiply $3$ by $- 8$ .

$\frac{- 35 - 40 y - 24 y}{3} = 8$

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

Step 2.2.1.7

Subtract $24 y$ from $- 40 y$ .

$\frac{- 35 - 64 y}{3} = 8$

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

Step 2.2.1.8

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- 1 \cdot 35 - 64 y}{3} = 8$

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

Step 2.2.1.9

Factor $- 1$ out of $- 64 y$ .

$\frac{- 1 \cdot 35 - (64 y)}{3} = 8$

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

Step 2.2.1.10

Factor $- 1$ out of $- 1 (35) - (64 y)$ .

$\frac{- 1 (35 + 64 y)}{3} = 8$

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

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{35 + 64 y}{3} = 8$

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

$- \frac{35 + 64 y}{3} = 8$

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

$- \frac{35 + 64 y}{3} = 8$

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

$- \frac{35 + 64 y}{3} = 8$

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

Step 3

Solve for $y$ in $- \frac{35 + 64 y}{3} = 8$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{35 + 64 y}{3}) = - 3 \cdot 8$

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

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 $- 3 (- \frac{35 + 64 y}{3})$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{35 + 64 y}{3}$ into the numerator.

$- 3 \frac{- (35 + 64 y)}{3} = - 3 \cdot 8$

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

Step 3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (35 + 64 y)}{3}) = - 3 \cdot 8$

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

Step 3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (35 + 64 y)}{3}) = - 3 \cdot 8$

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

Step 3.2.1.1.1.4

Rewrite the expression.

$35 + 64 y = - 3 \cdot 8$

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

$35 + 64 y = - 3 \cdot 8$

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

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (35 + 64 y) = - 3 \cdot 8$

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

Step 3.2.1.1.2.2

Multiply $35 + 64 y$ by $1$ .

$35 + 64 y = - 3 \cdot 8$

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

$35 + 64 y = - 3 \cdot 8$

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

$35 + 64 y = - 3 \cdot 8$

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

$35 + 64 y = - 3 \cdot 8$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 3$ by $8$ .

$35 + 64 y = - 24$

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

$35 + 64 y = - 24$

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

$35 + 64 y = - 24$

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

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 $35$ from both sides of the equation.

$64 y = - 24 - 35$

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

Step 3.3.2

Subtract $35$ from $- 24$ .

$64 y = - 59$

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

$64 y = - 59$

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

Step 3.4

Divide each term in $64 y = - 59$ by $64$ and simplify.

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Step 3.4.1

Divide each term in $64 y = - 59$ by $64$ .

$\frac{64 y}{64} = \frac{- 59}{64}$

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $64$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{64 y}{64} = \frac{- 59}{64}$

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 59}{64}$

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

$y = \frac{- 59}{64}$

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

$y = \frac{- 59}{64}$

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$y = - \frac{59}{64}$

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

$y = - \frac{59}{64}$

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

$y = - \frac{59}{64}$

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

$y = - \frac{59}{64}$

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

Step 4

Replace all occurrences of $y$ with $- \frac{59}{64}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{7}{3} - \frac{8 y}{3}$ with $- \frac{59}{64}$ .

$x = - \frac{7}{3} - \frac{8 (- \frac{59}{64})}{3}$

$y = - \frac{59}{64}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{7}{3} - \frac{8 (- \frac{59}{64})}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 7 - 8 (- \frac{59}{64})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.2.1.1

Move the leading negative in $- \frac{59}{64}$ into the numerator.

$x = \frac{- 7 - 8 (\frac{- 59}{64})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.1.2

Factor $8$ out of $- 8$ .

$x = \frac{- 7 + 8 (- 1) (\frac{- 59}{64})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.1.3

Factor $8$ out of $64$ .

$x = \frac{- 7 + 8 \cdot (- 1 \frac{- 59}{8 \cdot 8})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.1.4

Cancel the common factor.

$x = \frac{- 7 + 8 \cdot (- 1 \frac{- 59}{8 \cdot 8})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.1.5

Rewrite the expression.

$x = \frac{- 7 - 1 (\frac{- 59}{8})}{3}$

$y = - \frac{59}{64}$

$x = \frac{- 7 - 1 (\frac{- 59}{8})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = \frac{- 7 - 1 (- \frac{59}{8})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.3

Multiply $- 1 (- \frac{59}{8})$ .

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Step 4.2.1.2.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- 7 + 1 (\frac{59}{8})}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.2.3.2

Multiply $\frac{59}{8}$ by $1$ .

$x = \frac{- 7 + \frac{59}{8}}{3}$

$y = - \frac{59}{64}$

$x = \frac{- 7 + \frac{59}{8}}{3}$

$y = - \frac{59}{64}$

$x = \frac{- 7 + \frac{59}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.3

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x = \frac{- 7 \cdot \frac{8}{8} + \frac{59}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.4

Combine $- 7$ and $\frac{8}{8}$ .

$x = \frac{\frac{- 7 \cdot 8}{8} + \frac{59}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 7 \cdot 8 + 59}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 7$ by $8$ .

$x = \frac{\frac{- 56 + 59}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.6.2

Add $- 56$ and $59$ .

$x = \frac{\frac{3}{8}}{3}$

$y = - \frac{59}{64}$

$x = \frac{\frac{3}{8}}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3}{8} \cdot \frac{1}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.8

Cancel the common factor of $3$ .

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Step 4.2.1.8.1

Cancel the common factor.

$x = \frac{3}{8} \cdot \frac{1}{3}$

$y = - \frac{59}{64}$

Step 4.2.1.8.2

Rewrite the expression.

$x = \frac{1}{8}$

$y = - \frac{59}{64}$

$x = \frac{1}{8}$

$y = - \frac{59}{64}$

$x = \frac{1}{8}$

$y = - \frac{59}{64}$

$x = \frac{1}{8}$

$y = - \frac{59}{64}$

$x = \frac{1}{8}$

$y = - \frac{59}{64}$

Step 5

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

$(\frac{1}{8}, - \frac{59}{64})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{8}, - \frac{59}{64})$

Equation Form:

$x = \frac{1}{8}, y = - \frac{59}{64}$

Step 7