Finite Math Examples

$(- 2 \frac{3}{4}, 5)$ , $(1 \frac{3}{4}, 3)$

Step 1

Find the slope of the line between $(- 2 \frac{3}{4}, 5)$ and $(1 \frac{3}{4}, 3)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{3 - (5)}{1 \frac{3}{4} - (- 2 \frac{3}{4})}$

Step 1.4

Simplify.

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Step 1.4.1

Convert $1 \frac{3}{4}$ to an improper fraction.

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Step 1.4.1.1

A mixed number is an addition of its whole and fractional parts.

$m = \frac{3 - (5)}{1 + \frac{3}{4} - (- 2 \frac{3}{4})}$

Step 1.4.1.2

Add $1$ and $\frac{3}{4}$ .

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Step 1.4.1.2.1

Write $1$ as a fraction with a common denominator.

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

Step 1.4.1.2.2

Combine the numerators over the common denominator.

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

Step 1.4.1.2.3

Add $4$ and $3$ .

$m = \frac{3 - (5)}{\frac{7}{4} - (- 2 \frac{3}{4})}$

Step 1.4.2

Convert $2 \frac{3}{4}$ to an improper fraction.

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Step 1.4.2.1

A mixed number is an addition of its whole and fractional parts.

$m = \frac{3 - (5)}{\frac{7}{4} + 2 + \frac{3}{4}}$

Step 1.4.2.2

Add $2$ and $\frac{3}{4}$ .

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Step 1.4.2.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$m = \frac{3 - (5)}{\frac{7}{4} + 2 \cdot \frac{4}{4} + \frac{3}{4}}$

Step 1.4.2.2.2

Combine $2$ and $\frac{4}{4}$ .

$m = \frac{3 - (5)}{\frac{7}{4} + \frac{2 \cdot 4}{4} + \frac{3}{4}}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$m = \frac{3 - (5)}{\frac{7}{4} + \frac{2 \cdot 4 + 3}{4}}$

Step 1.4.2.2.4

Simplify the numerator.

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Step 1.4.2.2.4.1

Multiply $2$ by $4$ .

$m = \frac{3 - (5)}{\frac{7}{4} + \frac{8 + 3}{4}}$

Step 1.4.2.2.4.2

Add $8$ and $3$ .

$m = \frac{3 - (5)}{\frac{7}{4} + \frac{11}{4}}$

Step 1.4.3

Multiply the numerator and denominator of the fraction by $4$ .

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Step 1.4.3.1

Multiply $\frac{3 - (5)}{\frac{7}{4} - - \frac{11}{4}}$ by $\frac{4}{4}$ .

$m = \frac{4}{4} \cdot \frac{3 - (5)}{\frac{7}{4} + \frac{11}{4}}$

Step 1.4.3.2

Combine.

$m = \frac{4 (3 - (5))}{4 (\frac{7}{4} + \frac{11}{4})}$

Step 1.4.4

Apply the distributive property.

$m = \frac{4 \cdot 3 + 4 (- (5))}{4 (\frac{7}{4}) + 4 (\frac{11}{4})}$

Step 1.4.5

Cancel the common factor of $4$ .

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Step 1.4.5.1

Cancel the common factor.

$m = \frac{4 \cdot 3 + 4 (- (5))}{4 (\frac{7}{4}) + 4 (\frac{11}{4})}$

Step 1.4.5.2

Rewrite the expression.

$m = \frac{4 \cdot 3 + 4 (- (5))}{7 + 4 (\frac{11}{4})}$

Step 1.4.6

Simplify the numerator.

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Step 1.4.6.1

Multiply $4$ by $3$ .

$m = \frac{12 + 4 (- (5))}{7 + 4 (\frac{11}{4})}$

Step 1.4.6.2

Multiply $4 (- (5))$ .

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Step 1.4.6.2.1

Multiply $- 1$ by $5$ .

$m = \frac{12 + 4 \cdot - 5}{7 + 4 (\frac{11}{4})}$

Step 1.4.6.2.2

Multiply $4$ by $- 5$ .

$m = \frac{12 - 20}{7 + 4 (\frac{11}{4})}$

Step 1.4.6.3

Subtract $20$ from $12$ .

$m = \frac{- 8}{7 + 4 (\frac{11}{4})}$

Step 1.4.7

Simplify the denominator.

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Step 1.4.7.1

Multiply $- - \frac{11}{4}$ .

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Step 1.4.7.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 8}{7 + 4 (1 (\frac{11}{4}))}$

Step 1.4.7.1.2

Multiply $\frac{11}{4}$ by $1$ .

$m = \frac{- 8}{7 + 4 (\frac{11}{4})}$

Step 1.4.7.2

Cancel the common factor of $4$ .

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Step 1.4.7.2.1

Cancel the common factor.

$m = \frac{- 8}{7 + 4 (\frac{11}{4})}$

Step 1.4.7.2.2

Rewrite the expression.

$m = \frac{- 8}{7 + 11}$

Step 1.4.7.3

Add $7$ and $11$ .

$m = \frac{- 8}{18}$

Step 1.4.8

Reduce the expression by cancelling the common factors.

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Step 1.4.8.1

Cancel the common factor of $- 8$ and $18$ .

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Step 1.4.8.1.1

Factor $2$ out of $- 8$ .

$m = \frac{2 (- 4)}{18}$

Step 1.4.8.1.2

Cancel the common factors.

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Step 1.4.8.1.2.1

Factor $2$ out of $18$ .

$m = \frac{2 \cdot - 4}{2 \cdot 9}$

Step 1.4.8.1.2.2

Cancel the common factor.

$m = \frac{2 \cdot - 4}{2 \cdot 9}$

Step 1.4.8.1.2.3

Rewrite the expression.

$m = \frac{- 4}{9}$

Step 1.4.8.2

Move the negative in front of the fraction.

$m = - \frac{4}{9}$

Step 2

Use the slope $- \frac{4}{9}$ and a given point $(- 2 \frac{3}{4}, 5)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

Simplify the equation and keep it in point-slope form.

$y - 5 = - \frac{4}{9} \cdot (x + \frac{11}{4})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{4}{9} \cdot (x + \frac{11}{4})$ .

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Step 4.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{4}{9} \cdot (x + \frac{11}{4})$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Apply the distributive property.

$y - 5 = - \frac{4}{9} x - \frac{4}{9} \cdot \frac{11}{4}$

Step 4.1.2.2

Combine $x$ and $\frac{4}{9}$ .

$y - 5 = - \frac{x \cdot 4}{9} - \frac{4}{9} \cdot \frac{11}{4}$

Step 4.1.2.3

Cancel the common factor of $4$ .

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Step 4.1.2.3.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$y - 5 = - \frac{x \cdot 4}{9} + \frac{- 4}{9} \cdot \frac{11}{4}$

Step 4.1.2.3.2

Factor $4$ out of $- 4$ .

$y - 5 = - \frac{x \cdot 4}{9} + \frac{4 (- 1)}{9} \cdot \frac{11}{4}$

Step 4.1.2.3.3

Cancel the common factor.

$y - 5 = - \frac{x \cdot 4}{9} + \frac{4 \cdot - 1}{9} \cdot \frac{11}{4}$

Step 4.1.2.3.4

Rewrite the expression.

$y - 5 = - \frac{x \cdot 4}{9} + \frac{- 1}{9} \cdot 11$

Step 4.1.2.4

Combine $\frac{- 1}{9}$ and $11$ .

$y - 5 = - \frac{x \cdot 4}{9} + \frac{- 1 \cdot 11}{9}$

Step 4.1.2.5

Multiply $- 1$ by $11$ .

$y - 5 = - \frac{x \cdot 4}{9} + \frac{- 11}{9}$

Step 4.1.3

Simplify each term.

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Step 4.1.3.1

Move $4$ to the left of $x$ .

$y - 5 = - \frac{4 \cdot x}{9} + \frac{- 11}{9}$

Step 4.1.3.2

Move the negative in front of the fraction.

$y - 5 = - \frac{4 x}{9} - \frac{11}{9}$

Step 4.2

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

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Step 4.2.1

Add $5$ to both sides of the equation.

$y = - \frac{4 x}{9} - \frac{11}{9} + 5$

Step 4.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = - \frac{4 x}{9} - \frac{11}{9} + 5 \cdot \frac{9}{9}$

Step 4.2.3

Combine $5$ and $\frac{9}{9}$ .

$y = - \frac{4 x}{9} - \frac{11}{9} + \frac{5 \cdot 9}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{4 x}{9} + \frac{- 11 + 5 \cdot 9}{9}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $5$ by $9$ .

$y = - \frac{4 x}{9} + \frac{- 11 + 45}{9}$

Step 4.2.5.2

Add $- 11$ and $45$ .

$y = - \frac{4 x}{9} + \frac{34}{9}$

Step 5

Reorder terms.

$y = - (\frac{4}{9} x) + \frac{34}{9}$

Step 6

Remove parentheses.

$y = - \frac{4}{9} x + \frac{34}{9}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{4}{9} x + \frac{34}{9}$

Point-slope form:

$y - 5 = - \frac{4}{9} \cdot (x + \frac{11}{4})$

Step 8