Find the Equation Using Point-Slope Formula (1,-3) , (-7,2)

$(1, - 3)$ , $(- 7, 2)$

Step 1

Find the slope of the line between $(1, - 3)$ and $(- 7, 2)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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

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

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).

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

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

$m = \frac{2 - (- 3)}{- 7 - (1)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 3$ .

$m = \frac{2 + 3}{- 7 - (1)}$

Add $2$ and $3$ .

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

Simplify the denominator.

Multiply $- 1$ by $1$ .

$m = \frac{5}{- 7 - 1}$

Subtract $1$ from $- 7$ .

$m = \frac{5}{- 8}$

Move the negative in front of the fraction.

$m = - \frac{5}{8}$

Step 2

Use the slope $- \frac{5}{8}$ and a given point $(1, - 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 3) = - \frac{5}{8} \cdot (x - (1))$

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $- \frac{5}{8} \cdot (x - 1)$ .

Rewrite.

$y + 3 = 0 + 0 - \frac{5}{8} \cdot (x - 1)$

Simplify by adding zeros.

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

Apply the distributive property.

$y + 3 = - \frac{5}{8} x - \frac{5}{8} \cdot - 1$

Combine $x$ and $\frac{5}{8}$ .

$y + 3 = - \frac{x \cdot 5}{8} - \frac{5}{8} \cdot - 1$

Multiply $- \frac{5}{8} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{5}{8}$ by $1$ .

$y + 3 = - \frac{x \cdot 5}{8} + \frac{5}{8}$

Move $5$ to the left of $x$ .

$y + 3 = - \frac{5 x}{8} + \frac{5}{8}$

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

Subtract $3$ from both sides of the equation.

$y = - \frac{5 x}{8} + \frac{5}{8} - 3$

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

$y = - \frac{5 x}{8} + \frac{5}{8} - 3 \cdot \frac{8}{8}$

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

$y = - \frac{5 x}{8} + \frac{5}{8} + \frac{- 3 \cdot 8}{8}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{8} + \frac{5 - 3 \cdot 8}{8}$

Simplify the numerator.

Multiply $- 3$ by $8$ .

$y = - \frac{5 x}{8} + \frac{5 - 24}{8}$

Subtract $24$ from $5$ .

$y = - \frac{5 x}{8} + \frac{- 19}{8}$

Move the negative in front of the fraction.

$y = - \frac{5 x}{8} - \frac{19}{8}$

Step 5

Reorder terms.

$y = - (\frac{5}{8} x) - \frac{19}{8}$

Step 6

Remove parentheses.

$y = - \frac{5}{8} x - \frac{19}{8}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{5}{8} x - \frac{19}{8}$

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (1,-3) , (-7,2)

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