Find the Equation Using Point-Slope Formula (-1,5) , (3,6)

$(- 1, 5)$ , $(3, 6)$

Step 1

Find the slope of the line between $(- 1, 5)$ and $(3, 6)$ 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.

$m = \frac{6 - (5)}{3 - (- 1)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{6 - 5}{3 - (- 1)}$

Subtract $5$ from $6$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

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

Add $3$ and $1$ .

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

Step 2

Use the slope $\frac{1}{4}$ and a given point $(- 1, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = \frac{1}{4} \cdot (x - (- 1))$

Step 3

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

$y - 5 = \frac{1}{4} \cdot (x + 1)$

Step 4

Solve for $y$ .

Simplify $\frac{1}{4} \cdot (x + 1)$ .

Rewrite.

$y - 5 = 0 + 0 + \frac{1}{4} \cdot (x + 1)$

Simplify by adding zeros.

$y - 5 = \frac{1}{4} \cdot (x + 1)$

Apply the distributive property.

$y - 5 = \frac{1}{4} x + \frac{1}{4} \cdot 1$

Combine $\frac{1}{4}$ and $x$ .

$y - 5 = \frac{x}{4} + \frac{1}{4} \cdot 1$

Multiply $\frac{1}{4}$ by $1$ .

$y - 5 = \frac{x}{4} + \frac{1}{4}$

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

Add $5$ to both sides of the equation.

$y = \frac{x}{4} + \frac{1}{4} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x}{4} + \frac{1}{4} + 5 \cdot \frac{4}{4}$

Combine $5$ and $\frac{4}{4}$ .

$y = \frac{x}{4} + \frac{1}{4} + \frac{5 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{1 + 5 \cdot 4}{4}$

Simplify the numerator.

Multiply $5$ by $4$ .

$y = \frac{x}{4} + \frac{1 + 20}{4}$

Add $1$ and $20$ .

$y = \frac{x}{4} + \frac{21}{4}$

Step 5

Reorder terms.

$y = \frac{1}{4} x + \frac{21}{4}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{1}{4} x + \frac{21}{4}$

Point-slope form:

$y - 5 = \frac{1}{4} \cdot (x + 1)$

Step 7

Find the Equation Using Point-Slope Formula (-1,5) , (3,6)

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