Find the Equation Using Point-Slope Formula (4,3) , (1,5)

$(4, 3)$ , $(1, 5)$

Step 1

Find the slope of the line between $(4, 3)$ and $(1, 5)$ 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.

$m = \frac{5 - (3)}{1 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{5 - 3}{1 - (4)}$

Subtract $3$ from $5$ .

$m = \frac{2}{1 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{2}{1 - 4}$

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Multiply $- \frac{2}{3} \cdot - 4$ .

Multiply $- 4$ by $- 1$ .

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

Combine $4$ and $\frac{2}{3}$ .

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

Multiply $4$ by $2$ .

$y - 3 = - \frac{x \cdot 2}{3} + \frac{8}{3}$

Move $2$ to the left of $x$ .

$y - 3 = - \frac{2 x}{3} + \frac{8}{3}$

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

Add $3$ to both sides of the equation.

$y = - \frac{2 x}{3} + \frac{8}{3} + 3$

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

$y = - \frac{2 x}{3} + \frac{8}{3} + 3 \cdot \frac{3}{3}$

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

$y = - \frac{2 x}{3} + \frac{8}{3} + \frac{3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{2 x}{3} + \frac{8 + 3 \cdot 3}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$y = - \frac{2 x}{3} + \frac{8 + 9}{3}$

Add $8$ and $9$ .

$y = - \frac{2 x}{3} + \frac{17}{3}$

Step 5

Reorder terms.

$y = - (\frac{2}{3} x) + \frac{17}{3}$

Step 6

Remove parentheses.

$y = - \frac{2}{3} x + \frac{17}{3}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{2}{3} x + \frac{17}{3}$

Point-slope form:

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

Step 8

Find the Equation Using Point-Slope Formula (4,3) , (1,5)

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