Find the Equation Using Point-Slope Formula (6,-7) , (5,-10)

$(6, - 7)$ , $(5, - 10)$

Step 1

Find the slope of the line between $(6, - 7)$ and $(5, - 10)$ 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.

$m = \frac{- 10 - (- 7)}{5 - (6)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 7$ .

$m = \frac{- 10 + 7}{5 - (6)}$

Add $- 10$ and $7$ .

$m = \frac{- 3}{5 - (6)}$

Simplify the denominator.

Multiply $- 1$ by $6$ .

$m = \frac{- 3}{5 - 6}$

Subtract $6$ from $5$ .

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

Divide $- 3$ by $- 1$ .

$m = 3$

Step 2

Use the slope $3$ and a given point $(6, - 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 7) = 3 \cdot (x - (6))$

Step 3

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

$y + 7 = 3 \cdot (x - 6)$

Step 4

Solve for $y$ .

Simplify $3 \cdot (x - 6)$ .

Rewrite.

$y + 7 = 0 + 0 + 3 \cdot (x - 6)$

Simplify by adding zeros.

$y + 7 = 3 \cdot (x - 6)$

Apply the distributive property.

$y + 7 = 3 x + 3 \cdot - 6$

Multiply $3$ by $- 6$ .

$y + 7 = 3 x - 18$

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

Subtract $7$ from both sides of the equation.

$y = 3 x - 18 - 7$

Subtract $7$ from $- 18$ .

$y = 3 x - 25$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 3 x - 25$

Point-slope form:

$y + 7 = 3 \cdot (x - 6)$

Step 6

Find the Equation Using Point-Slope Formula (6,-7) , (5,-10)

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