Find the Equation Using Point-Slope Formula (5,15) , (10,20)

$(5, 15)$ , $(10, 20)$

Step 1

Find the slope of the line between $(5, 15)$ and $(10, 20)$ 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.

$m = \frac{20 - (15)}{10 - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $15$ .

$m = \frac{20 - 15}{10 - (5)}$

Subtract $15$ from $20$ .

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

Simplify the denominator.

Multiply $- 1$ by $5$ .

$m = \frac{5}{10 - 5}$

Subtract $5$ from $10$ .

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

Divide $5$ by $5$ .

$m = 1$

Step 2

Use the slope $1$ and a given point $(5, 15)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (15) = 1 \cdot (x - (5))$

Step 3

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

$y - 15 = 1 \cdot (x - 5)$

Step 4

Solve for $y$ .

Multiply $x - 5$ by $1$ .

$y - 15 = x - 5$

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

Add $15$ to both sides of the equation.

$y = x - 5 + 15$

Add $- 5$ and $15$ .

$y = x + 10$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = x + 10$

Point-slope form:

$y - 15 = 1 \cdot (x - 5)$

Step 6

Find the Equation Using Point-Slope Formula (5,15) , (10,20)

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