Find the Equation Using Point-Slope Formula (2,5) , (4,8)

$(2, 5)$ , $(4, 8)$

Step 1

Find the slope of the line between $(2, 5)$ and $(4, 8)$ 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.

$m = \frac{8 - (5)}{4 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{8 - 5}{4 - (2)}$

Subtract $5$ from $8$ .

$m = \frac{3}{4 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{3}{4 - 2}$

Subtract $2$ from $4$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{3}{2}$ and $x$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $- 1$ .

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

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

Add $5$ to both sides of the equation.

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

Add $- 3$ and $5$ .

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (2,5) , (4,8)

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