Find the Equation Using Point-Slope Formula (2,3) and (5,4)

$(2, 3)$ and $(5, 4)$

Step 1

Find the slope of the line between $(2, 3)$ and $(5, 4)$ 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.

$m = \frac{4 - (3)}{5 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{4 - 3}{5 - (2)}$

Subtract $3$ from $4$ .

$m = \frac{1}{5 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{1}{5 - 2}$

Subtract $2$ from $5$ .

$m = \frac{1}{3}$

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

Simplify $\frac{1}{3} \cdot (x - 2)$ .

Rewrite.

$y - 3 = 0 + 0 + \frac{1}{3} \cdot (x - 2)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{1}{3}$ and $x$ .

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

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

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

Move the negative in front of the fraction.

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

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

Add $3$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

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

Add $- 2$ and $9$ .

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

Step 5

Reorder terms.

$y = \frac{1}{3} x + \frac{7}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{3} x + \frac{7}{3}$

Point-slope form:

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

Step 7

Find the Equation Using Point-Slope Formula (2,3) and (5,4)

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