Precalculus Examples

$(- 7, 1)$ , $(8, - 2)$

Step 1

Find the slope of the line between $(- 7, 1)$ and $(8, - 2)$ 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.

$m = \frac{- 2 - (1)}{8 - (- 7)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $1$ .

$m = \frac{- 2 - 1}{8 - (- 7)}$

Step 1.4.1.2

Subtract $1$ from $- 2$ .

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

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 7$ .

$m = \frac{- 3}{8 + 7}$

Step 1.4.2.2

Add $8$ and $7$ .

$m = \frac{- 3}{15}$

Step 1.4.3

Reduce the expression by cancelling the common factors.

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Step 1.4.3.1

Cancel the common factor of $- 3$ and $15$ .

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Step 1.4.3.1.1

Factor $3$ out of $- 3$ .

$m = \frac{3 (- 1)}{15}$

Step 1.4.3.1.2

Cancel the common factors.

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Step 1.4.3.1.2.1

Factor $3$ out of $15$ .

$m = \frac{3 \cdot - 1}{3 \cdot 5}$

Step 1.4.3.1.2.2

Cancel the common factor.

$m = \frac{3 \cdot - 1}{3 \cdot 5}$

Step 1.4.3.1.2.3

Rewrite the expression.

$m = \frac{- 1}{5}$

Step 1.4.3.2

Move the negative in front of the fraction.

$m = - \frac{1}{5}$

Step 2

Use the slope $- \frac{1}{5}$ and a given point $(- 7, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{5} \cdot (x - (- 7))$

Step 3

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

$y - 1 = - \frac{1}{5} \cdot (x + 7)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{1}{5} \cdot (x + 7)$ .

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Step 4.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{5} \cdot (x + 7)$

Step 4.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{5} \cdot (x + 7)$

Step 4.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{5} x - \frac{1}{5} \cdot 7$

Step 4.1.4

Combine $x$ and $\frac{1}{5}$ .

$y - 1 = - \frac{x}{5} - \frac{1}{5} \cdot 7$

Step 4.1.5

Multiply $- \frac{1}{5} \cdot 7$ .

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Step 4.1.5.1

Multiply $7$ by $- 1$ .

$y - 1 = - \frac{x}{5} - 7 (\frac{1}{5})$

Step 4.1.5.2

Combine $- 7$ and $\frac{1}{5}$ .

$y - 1 = - \frac{x}{5} + \frac{- 7}{5}$

Step 4.1.6

Move the negative in front of the fraction.

$y - 1 = - \frac{x}{5} - \frac{7}{5}$

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 $1$ to both sides of the equation.

$y = - \frac{x}{5} - \frac{7}{5} + 1$

Step 4.2.2

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

$y = - \frac{x}{5} - \frac{7}{5} + \frac{5}{5}$

Step 4.2.3

Combine the numerators over the common denominator.

$y = - \frac{x}{5} + \frac{- 7 + 5}{5}$

Step 4.2.4

Add $- 7$ and $5$ .

$y = - \frac{x}{5} + \frac{- 2}{5}$

Step 4.2.5

Move the negative in front of the fraction.

$y = - \frac{x}{5} - \frac{2}{5}$

Step 5

Reorder terms.

$y = - (\frac{1}{5} x) - \frac{2}{5}$

Step 6

Remove parentheses.

$y = - \frac{1}{5} x - \frac{2}{5}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{5} x - \frac{2}{5}$

Point-slope form:

$y - 1 = - \frac{1}{5} \cdot (x + 7)$

Step 8