Find the Equation Using Point-Slope Formula (1,4) , (4,13)

$(1, 4)$ , $(4, 13)$

Step 1

Find the slope of the line between $(1, 4)$ and $(4, 13)$ 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.

$m = \frac{13 - (4)}{4 - (1)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $4$ .

$m = \frac{13 - 4}{4 - (1)}$

Subtract $4$ from $13$ .

$m = \frac{9}{4 - (1)}$

Simplify the denominator.

Multiply $- 1$ by $1$ .

$m = \frac{9}{4 - 1}$

Subtract $1$ from $4$ .

$m = \frac{9}{3}$

Divide $9$ by $3$ .

$m = 3$

Step 2

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

$y - (4) = 3 \cdot (x - (1))$

Step 3

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

$y - 4 = 3 \cdot (x - 1)$

Step 4

Solve for $y$ .

Simplify $3 \cdot (x - 1)$ .

Rewrite.

$y - 4 = 0 + 0 + 3 \cdot (x - 1)$

Simplify by adding zeros.

$y - 4 = 3 \cdot (x - 1)$

Apply the distributive property.

$y - 4 = 3 x + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$y - 4 = 3 x - 3$

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

Add $4$ to both sides of the equation.

$y = 3 x - 3 + 4$

Add $- 3$ and $4$ .

$y = 3 x + 1$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 3 x + 1$

Point-slope form:

$y - 4 = 3 \cdot (x - 1)$

Step 6

Find the Equation Using Point-Slope Formula (1,4) , (4,13)

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