Find the Equation Using Point-Slope Formula (2,-5) , (-1,4)

$(2, - 5)$ , $(- 1, 4)$

Step 1

Find the slope of the line between $(2, - 5)$ and $(- 1, 4)$ 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.

$m = \frac{4 - (- 5)}{- 1 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $- 5$ .

$m = \frac{4 + 5}{- 1 - (2)}$

Add $4$ and $5$ .

$m = \frac{9}{- 1 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{9}{- 1 - 2}$

Subtract $2$ from $- 1$ .

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

Divide $9$ by $- 3$ .

$m = - 3$

Step 2

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

$y - (- 5) = - 3 \cdot (x - (2))$

Step 3

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

$y + 5 = - 3 \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $- 3 \cdot (x - 2)$ .

Rewrite.

$y + 5 = 0 + 0 - 3 \cdot (x - 2)$

Simplify by adding zeros.

$y + 5 = - 3 \cdot (x - 2)$

Apply the distributive property.

$y + 5 = - 3 x - 3 \cdot - 2$

Multiply $- 3$ by $- 2$ .

$y + 5 = - 3 x + 6$

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

Subtract $5$ from both sides of the equation.

$y = - 3 x + 6 - 5$

Subtract $5$ from $6$ .

$y = - 3 x + 1$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = - 3 x + 1$

Point-slope form:

$y + 5 = - 3 \cdot (x - 2)$

Step 6

Find the Equation Using Point-Slope Formula (2,-5) , (-1,4)

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