Precalculus Examples

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

Step 1

Find the slope of the line between $(\frac{3}{4}, \frac{3}{2})$ and $(- \frac{4}{3}, \frac{7}{4})$ 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.

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

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $12$ .

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Step 1.4.1.1

Multiply $\frac{\frac{7}{4} - \frac{3}{2}}{- \frac{4}{3} - \frac{3}{4}}$ by $\frac{12}{12}$ .

$m = \frac{12}{12} \cdot \frac{\frac{7}{4} - \frac{3}{2}}{- \frac{4}{3} - \frac{3}{4}}$

Step 1.4.1.2

Combine.

$m = \frac{12 (\frac{7}{4} - \frac{3}{2})}{12 (- \frac{4}{3} - \frac{3}{4})}$

Step 1.4.2

Apply the distributive property.

$m = \frac{12 (\frac{7}{4}) + 12 (- \frac{3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3

Simplify by cancelling.

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Step 1.4.3.1

Cancel the common factor of $4$ .

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Step 1.4.3.1.1

Factor $4$ out of $12$ .

$m = \frac{4 (3) (\frac{7}{4}) + 12 (- \frac{3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.1.2

Cancel the common factor.

$m = \frac{4 \cdot (3 (\frac{7}{4})) + 12 (- \frac{3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.1.3

Rewrite the expression.

$m = \frac{3 \cdot 7 + 12 (- \frac{3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.2

Multiply $3$ by $7$ .

$m = \frac{21 + 12 (- \frac{3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.3

Cancel the common factor of $2$ .

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Step 1.4.3.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$m = \frac{21 + 12 (\frac{- 3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.3.2

Factor $2$ out of $12$ .

$m = \frac{21 + 2 (6) (\frac{- 3}{2})}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.3.3

Cancel the common factor.

$m = \frac{21 + 2 \cdot (6 (\frac{- 3}{2}))}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.3.4

Rewrite the expression.

$m = \frac{21 + 6 \cdot - 3}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.4

Multiply $6$ by $- 3$ .

$m = \frac{21 - 18}{12 (- \frac{4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.5

Cancel the common factor of $3$ .

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Step 1.4.3.5.1

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

$m = \frac{21 - 18}{12 (\frac{- 4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.5.2

Factor $3$ out of $12$ .

$m = \frac{21 - 18}{3 (4) (\frac{- 4}{3}) + 12 (- \frac{3}{4})}$

Step 1.4.3.5.3

Cancel the common factor.

$m = \frac{21 - 18}{3 \cdot (4 (\frac{- 4}{3})) + 12 (- \frac{3}{4})}$

Step 1.4.3.5.4

Rewrite the expression.

$m = \frac{21 - 18}{4 \cdot - 4 + 12 (- \frac{3}{4})}$

Step 1.4.3.6

Multiply $4$ by $- 4$ .

$m = \frac{21 - 18}{- 16 + 12 (- \frac{3}{4})}$

Step 1.4.3.7

Cancel the common factor of $4$ .

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Step 1.4.3.7.1

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

$m = \frac{21 - 18}{- 16 + 12 (\frac{- 3}{4})}$

Step 1.4.3.7.2

Factor $4$ out of $12$ .

$m = \frac{21 - 18}{- 16 + 4 (3) (\frac{- 3}{4})}$

Step 1.4.3.7.3

Cancel the common factor.

$m = \frac{21 - 18}{- 16 + 4 \cdot (3 (\frac{- 3}{4}))}$

Step 1.4.3.7.4

Rewrite the expression.

$m = \frac{21 - 18}{- 16 + 3 \cdot - 3}$

Step 1.4.3.8

Multiply $3$ by $- 3$ .

$m = \frac{21 - 18}{- 16 - 9}$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Subtract $18$ from $21$ .

$m = \frac{3}{- 16 - 9}$

Step 1.4.4.2

Subtract $9$ from $- 16$ .

$m = \frac{3}{- 25}$

Step 1.4.4.3

Move the negative in front of the fraction.

$m = - \frac{3}{25}$

Step 2

Use the slope $- \frac{3}{25}$ and a given point $(\frac{3}{4}, \frac{3}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{3}{2}) = - \frac{3}{25} \cdot (x - (\frac{3}{4}))$

Step 3

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

$y - \frac{3}{2} = - \frac{3}{25} \cdot (x - \frac{3}{4})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{3}{25} \cdot (x - \frac{3}{4})$ .

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Step 4.1.1

Rewrite.

$y - \frac{3}{2} = 0 + 0 - \frac{3}{25} \cdot (x - \frac{3}{4})$

Step 4.1.2

Simplify by adding zeros.

$y - \frac{3}{2} = - \frac{3}{25} \cdot (x - \frac{3}{4})$

Step 4.1.3

Apply the distributive property.

$y - \frac{3}{2} = - \frac{3}{25} x - \frac{3}{25} (- \frac{3}{4})$

Step 4.1.4

Combine $x$ and $\frac{3}{25}$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} - \frac{3}{25} (- \frac{3}{4})$

Step 4.1.5

Multiply $- \frac{3}{25} (- \frac{3}{4})$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} + 1 (\frac{3}{25}) \frac{3}{4}$

Step 4.1.5.2

Multiply $\frac{3}{25}$ by $1$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} + \frac{3}{25} \cdot \frac{3}{4}$

Step 4.1.5.3

Multiply $\frac{3}{25}$ by $\frac{3}{4}$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} + \frac{3 \cdot 3}{25 \cdot 4}$

Step 4.1.5.4

Multiply $3$ by $3$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} + \frac{9}{25 \cdot 4}$

Step 4.1.5.5

Multiply $25$ by $4$ .

$y - \frac{3}{2} = - \frac{x \cdot 3}{25} + \frac{9}{100}$

Step 4.1.6

Move $3$ to the left of $x$ .

$y - \frac{3}{2} = - \frac{3 x}{25} + \frac{9}{100}$

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 $\frac{3}{2}$ to both sides of the equation.

$y = - \frac{3 x}{25} + \frac{9}{100} + \frac{3}{2}$

Step 4.2.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{50}{50}$ .

$y = - \frac{3 x}{25} + \frac{9}{100} + \frac{3}{2} \cdot \frac{50}{50}$

Step 4.2.3

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{3}{2}$ by $\frac{50}{50}$ .

$y = - \frac{3 x}{25} + \frac{9}{100} + \frac{3 \cdot 50}{2 \cdot 50}$

Step 4.2.3.2

Multiply $2$ by $50$ .

$y = - \frac{3 x}{25} + \frac{9}{100} + \frac{3 \cdot 50}{100}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{25} + \frac{9 + 3 \cdot 50}{100}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $3$ by $50$ .

$y = - \frac{3 x}{25} + \frac{9 + 150}{100}$

Step 4.2.5.2

Add $9$ and $150$ .

$y = - \frac{3 x}{25} + \frac{159}{100}$

Step 5

Reorder terms.

$y = - (\frac{3}{25} x) + \frac{159}{100}$

Step 6

Remove parentheses.

$y = - \frac{3}{25} x + \frac{159}{100}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{25} x + \frac{159}{100}$

Point-slope form:

$y - \frac{3}{2} = - \frac{3}{25} \cdot (x - \frac{3}{4})$

Step 8