Find the Equation Using Point-Slope Formula (-2,6) and (1,1)

$(- 2, 6)$ and $(1, 1)$

Find the slope of the line between $(- 2, 6)$ and $(1, 1)$ 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.

$m = \frac{1 - (6)}{1 - (- 2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $6$ .

$m = \frac{1 - 6}{1 - (- 2)}$

Subtract $6$ from $1$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

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

Add $1$ and $2$ .

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

Move the negative in front of the fraction.

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

Use the slope $- \frac{5}{3}$ and a given point $(- 2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - \frac{5}{3} \cdot (x - (- 2))$

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

$y - 6 = - \frac{5}{3} \cdot (x + 2)$

Solve for $y$ .

Simplify $- \frac{5}{3} \cdot (x + 2)$ .

Rewrite.

$y - 6 = 0 + 0 - \frac{5}{3} \cdot (x + 2)$

Simplify by adding zeros.

$y - 6 = - \frac{5}{3} \cdot (x + 2)$

Apply the distributive property.

$y - 6 = - \frac{5}{3} x - \frac{5}{3} \cdot 2$

Combine $x$ and $\frac{5}{3}$ .

$y - 6 = - \frac{x \cdot 5}{3} - \frac{5}{3} \cdot 2$

Multiply $- \frac{5}{3} \cdot 2$ .

Multiply $2$ by $- 1$ .

$y - 6 = - \frac{x \cdot 5}{3} - 2 (\frac{5}{3})$

Combine $- 2$ and $\frac{5}{3}$ .

$y - 6 = - \frac{x \cdot 5}{3} + \frac{- 2 \cdot 5}{3}$

Multiply $- 2$ by $5$ .

$y - 6 = - \frac{x \cdot 5}{3} + \frac{- 10}{3}$

Simplify each term.

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

$y - 6 = - \frac{5 \cdot x}{3} + \frac{- 10}{3}$

Move the negative in front of the fraction.

$y - 6 = - \frac{5 x}{3} - \frac{10}{3}$

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

Add $6$ to both sides of the equation.

$y = - \frac{5 x}{3} - \frac{10}{3} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{5 x}{3} - \frac{10}{3} + 6 \cdot \frac{3}{3}$

Combine $6$ and $\frac{3}{3}$ .

$y = - \frac{5 x}{3} - \frac{10}{3} + \frac{6 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{3} + \frac{- 10 + 6 \cdot 3}{3}$

Simplify the numerator.

Multiply $6$ by $3$ .

$y = - \frac{5 x}{3} + \frac{- 10 + 18}{3}$

Add $- 10$ and $18$ .

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

Reorder terms.

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

Remove parentheses.

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

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y - 6 = - \frac{5}{3} \cdot (x + 2)$

Find the Equation Using Point-Slope Formula (-2,6) and (1,1)

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