Find the Equation Using Point-Slope Formula (2,5) , (6,8)

$(2, 5)$ , $(6, 8)$

Step 1

Find the slope of the line between $(2, 5)$ and $(6, 8)$ 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.

$m = \frac{8 - (5)}{6 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{8 - 5}{6 - (2)}$

Subtract $5$ from $8$ .

$m = \frac{3}{6 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{3}{6 - 2}$

Subtract $2$ from $6$ .

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

Step 2

Use the slope $\frac{3}{4}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{3}{4} \cdot (x - (2))$

Step 3

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

$y - 5 = \frac{3}{4} \cdot (x - 2)$

Step 4

Solve for $y$ .

Simplify $\frac{3}{4} \cdot (x - 2)$ .

Rewrite.

$y - 5 = 0 + 0 + \frac{3}{4} \cdot (x - 2)$

Simplify by adding zeros.

$y - 5 = \frac{3}{4} \cdot (x - 2)$

Apply the distributive property.

$y - 5 = \frac{3}{4} x + \frac{3}{4} \cdot - 2$

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

$y - 5 = \frac{3 x}{4} + \frac{3}{4} \cdot - 2$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$y - 5 = \frac{3 x}{4} + \frac{3}{2 (2)} \cdot - 2$

Factor $2$ out of $- 2$ .

$y - 5 = \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1)$

Cancel the common factor.

$y - 5 = \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot (2 \cdot - 1)$

Rewrite the expression.

$y - 5 = \frac{3 x}{4} + \frac{3}{2} \cdot - 1$

Combine $\frac{3}{2}$ and $- 1$ .

$y - 5 = \frac{3 x}{4} + \frac{3 \cdot - 1}{2}$

Simplify the expression.

Multiply $3$ by $- 1$ .

$y - 5 = \frac{3 x}{4} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$y - 5 = \frac{3 x}{4} - \frac{3}{2}$

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

Add $5$ to both sides of the equation.

$y = \frac{3 x}{4} - \frac{3}{2} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3 x}{4} - \frac{3}{2} + 5 \cdot \frac{2}{2}$

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

$y = \frac{3 x}{4} - \frac{3}{2} + \frac{5 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{3 x}{4} + \frac{- 3 + 5 \cdot 2}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$y = \frac{3 x}{4} + \frac{- 3 + 10}{2}$

Add $- 3$ and $10$ .

$y = \frac{3 x}{4} + \frac{7}{2}$

Step 5

Reorder terms.

$y = \frac{3}{4} x + \frac{7}{2}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{3}{4} x + \frac{7}{2}$

Point-slope form:

$y - 5 = \frac{3}{4} \cdot (x - 2)$

Step 7

Find the Equation Using Point-Slope Formula (2,5) , (6,8)

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