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

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

Step 1

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

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

Simplify.

Simplify the numerator.

Multiply $- 1$ by $4$ .

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

Subtract $4$ from $5$ .

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

Simplify the denominator.

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $3$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $y$ .

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

Rewrite.

$y - 4 = 0 + 0 + \frac{1}{2} \cdot (x - 1)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $\frac{1}{2}$ and $x$ .

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

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

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

Move the negative in front of the fraction.

$y - 4 = \frac{x}{2} - \frac{1}{2}$

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

Add $4$ to both sides of the equation.

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $2$ .

$y = \frac{x}{2} + \frac{- 1 + 8}{2}$

Add $- 1$ and $8$ .

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

Step 5

Reorder terms.

$y = \frac{1}{2} x + \frac{7}{2}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{2} x + \frac{7}{2}$

Point-slope form:

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

Step 7

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

Related Questions

Support LCMGCF.com by disabling your adblocker.