Find the Equation Using Point-Slope Formula (4,-8) , (8,5)

$(4, - 8)$ , $(8, 5)$

Step 1

Find the slope of the line between $(4, - 8)$ and $(8, 5)$ 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.

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 8$ .

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

Add $5$ and $8$ .

$m = \frac{13}{8 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{13}{8 - 4}$

Subtract $4$ from $8$ .

$m = \frac{13}{4}$

Step 2

Use the slope $\frac{13}{4}$ and a given point $(4, - 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 8) = \frac{13}{4} \cdot (x - (4))$

Step 3

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

$y + 8 = \frac{13}{4} \cdot (x - 4)$

Step 4

Solve for $y$ .

Simplify $\frac{13}{4} \cdot (x - 4)$ .

Rewrite.

$y + 8 = 0 + 0 + \frac{13}{4} \cdot (x - 4)$

Simplify by adding zeros.

$y + 8 = \frac{13}{4} \cdot (x - 4)$

Apply the distributive property.

$y + 8 = \frac{13}{4} x + \frac{13}{4} \cdot - 4$

Combine $\frac{13}{4}$ and $x$ .

$y + 8 = \frac{13 x}{4} + \frac{13}{4} \cdot - 4$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

$y + 8 = \frac{13 x}{4} + \frac{13}{4} \cdot (4 (- 1))$

Cancel the common factor.

$y + 8 = \frac{13 x}{4} + \frac{13}{4} \cdot (4 \cdot - 1)$

Rewrite the expression.

$y + 8 = \frac{13 x}{4} + 13 \cdot - 1$

Multiply $13$ by $- 1$ .

$y + 8 = \frac{13 x}{4} - 13$

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

Subtract $8$ from both sides of the equation.

$y = \frac{13 x}{4} - 13 - 8$

Subtract $8$ from $- 13$ .

$y = \frac{13 x}{4} - 21$

Step 5

Reorder terms.

$y = \frac{13}{4} x - 21$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{13}{4} x - 21$

Point-slope form:

$y + 8 = \frac{13}{4} \cdot (x - 4)$

Step 7

Find the Equation Using Point-Slope Formula (4,-8) , (8,5)

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