Find the Equation Using Point-Slope Formula (20,150) , (60,120)

$(20, 150)$ , $(60, 120)$

Step 1

Find the slope of the line between $(20, 150)$ and $(60, 120)$ 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.

$m = \frac{120 - (150)}{60 - (20)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $150$ .

$m = \frac{120 - 150}{60 - (20)}$

Subtract $150$ from $120$ .

$m = \frac{- 30}{60 - (20)}$

Simplify the denominator.

Multiply $- 1$ by $20$ .

$m = \frac{- 30}{60 - 20}$

Subtract $20$ from $60$ .

$m = \frac{- 30}{40}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 30$ and $40$ .

Factor $10$ out of $- 30$ .

$m = \frac{10 (- 3)}{40}$

Cancel the common factors.

Factor $10$ out of $40$ .

$m = \frac{10 \cdot - 3}{10 \cdot 4}$

Cancel the common factor.

$m = \frac{10 \cdot - 3}{10 \cdot 4}$

Rewrite the expression.

$m = \frac{- 3}{4}$

Move the negative in front of the fraction.

$m = - \frac{3}{4}$

Step 2

Use the slope $- \frac{3}{4}$ and a given point $(20, 150)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (150) = - \frac{3}{4} \cdot (x - (20))$

Step 3

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

$y - 150 = - \frac{3}{4} \cdot (x - 20)$

Step 4

Solve for $y$ .

Simplify $- \frac{3}{4} \cdot (x - 20)$ .

Rewrite.

$y - 150 = 0 + 0 - \frac{3}{4} \cdot (x - 20)$

Simplify terms.

Apply the distributive property.

$y - 150 = - \frac{3}{4} x - \frac{3}{4} \cdot - 20$

Combine $x$ and $\frac{3}{4}$ .

$y - 150 = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot - 20$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y - 150 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot - 20$

Factor $4$ out of $- 20$ .

$y - 150 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot (4 (- 5))$

Cancel the common factor.

$y - 150 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot (4 \cdot - 5)$

Rewrite the expression.

$y - 150 = - \frac{x \cdot 3}{4} - 3 \cdot - 5$

Multiply $- 3$ by $- 5$ .

$y - 150 = - \frac{x \cdot 3}{4} + 15$

Move $3$ to the left of $x$ .

$y - 150 = - \frac{3 x}{4} + 15$

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

Add $150$ to both sides of the equation.

$y = - \frac{3 x}{4} + 15 + 150$

Add $15$ and $150$ .

$y = - \frac{3 x}{4} + 165$

Step 5

Reorder terms.

$y = - (\frac{3}{4} x) + 165$

Step 6

Remove parentheses.

$y = - \frac{3}{4} x + 165$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{4} x + 165$

Point-slope form:

$y - 150 = - \frac{3}{4} \cdot (x - 20)$

Step 8

Find the Equation Using Point-Slope Formula (20,150) , (60,120)

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