Finite Math Examples

$3 y = 2 x - 8$ , $5 x + 3 y = 1$

Step 1

Divide each term in $3 y = 2 x - 8$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 y = 2 x - 8$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 8}{3}$

$5 x + 3 y = 1$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 8}{3}$

$5 x + 3 y = 1$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{3} + \frac{- 8}{3}$

$5 x + 3 y = 1$

$y = \frac{2 x}{3} + \frac{- 8}{3}$

$5 x + 3 y = 1$

$y = \frac{2 x}{3} + \frac{- 8}{3}$

$5 x + 3 y = 1$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 3 y = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 3 y = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 3 y = 1$

Step 2

Replace all occurrences of $y$ with $\frac{2 x}{3} - \frac{8}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $5 x + 3 y = 1$ with $\frac{2 x}{3} - \frac{8}{3}$ .

$5 x + 3 (\frac{2 x}{3} - \frac{8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 x + 3 (\frac{2 x}{3} - \frac{8}{3})$ .

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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.

$5 x + 3 (\frac{2 x}{3}) + 3 (- \frac{8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$5 x + 3 (\frac{2 x}{3}) + 3 (- \frac{8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

$5 x + 2 x + 3 (- \frac{8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 2 x + 3 (- \frac{8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.1.1.3.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$5 x + 2 x + 3 (\frac{- 8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.1.3.2

Cancel the common factor.

$5 x + 2 x + 3 (\frac{- 8}{3}) = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.1.3.3

Rewrite the expression.

$5 x + 2 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 2 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$5 x + 2 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 2.2.1.2

Add $5 x$ and $2 x$ .

$7 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$7 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$7 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

$7 x - 8 = 1$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 3

Solve for $x$ in $7 x - 8 = 1$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Add $8$ to both sides of the equation.

$7 x = 1 + 8$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 3.1.2

Add $1$ and $8$ .

$7 x = 9$

$y = \frac{2 x}{3} - \frac{8}{3}$

$7 x = 9$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 3.2

Divide each term in $7 x = 9$ by $7$ and simplify.

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Step 3.2.1

Divide each term in $7 x = 9$ by $7$ .

$\frac{7 x}{7} = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

$x = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

$x = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

$x = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

$x = \frac{9}{7}$

$y = \frac{2 x}{3} - \frac{8}{3}$

Step 4

Replace all occurrences of $x$ with $\frac{9}{7}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{2 x}{3} - \frac{8}{3}$ with $\frac{9}{7}$ .

$y = \frac{2 (\frac{9}{7})}{3} - \frac{8}{3}$

$x = \frac{9}{7}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2 (\frac{9}{7})}{3} - \frac{8}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$y = \frac{2 (\frac{9}{7}) - 8}{3}$

$x = \frac{9}{7}$

Step 4.2.1.2

Multiply $2 (\frac{9}{7})$ .

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Step 4.2.1.2.1

Combine $2$ and $\frac{9}{7}$ .

$y = \frac{\frac{2 \cdot 9}{7} - 8}{3}$

$x = \frac{9}{7}$

Step 4.2.1.2.2

Multiply $2$ by $9$ .

$y = \frac{\frac{18}{7} - 8}{3}$

$x = \frac{9}{7}$

$y = \frac{\frac{18}{7} - 8}{3}$

$x = \frac{9}{7}$

Step 4.2.1.3

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

$y = \frac{\frac{18}{7} - 8 \cdot \frac{7}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.4

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

$y = \frac{\frac{18}{7} + \frac{- 8 \cdot 7}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$y = \frac{\frac{18 - 8 \cdot 7}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 8$ by $7$ .

$y = \frac{\frac{18 - 56}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.6.2

Subtract $56$ from $18$ .

$y = \frac{\frac{- 38}{7}}{3}$

$x = \frac{9}{7}$

$y = \frac{\frac{- 38}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.7

Move the negative in front of the fraction.

$y = \frac{- \frac{38}{7}}{3}$

$x = \frac{9}{7}$

Step 4.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{38}{7} \cdot \frac{1}{3}$

$x = \frac{9}{7}$

Step 4.2.1.9

Multiply $- \frac{38}{7} \cdot \frac{1}{3}$ .

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Step 4.2.1.9.1

Multiply $\frac{1}{3}$ by $\frac{38}{7}$ .

$y = - \frac{38}{3 \cdot 7}$

$x = \frac{9}{7}$

Step 4.2.1.9.2

Multiply $3$ by $7$ .

$y = - \frac{38}{21}$

$x = \frac{9}{7}$

$y = - \frac{38}{21}$

$x = \frac{9}{7}$

$y = - \frac{38}{21}$

$x = \frac{9}{7}$

$y = - \frac{38}{21}$

$x = \frac{9}{7}$

$y = - \frac{38}{21}$

$x = \frac{9}{7}$

Step 5

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

$(\frac{9}{7}, - \frac{38}{21})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{9}{7}, - \frac{38}{21})$

Equation Form:

$x = \frac{9}{7}, y = - \frac{38}{21}$

Step 7