Find the Equation Using Point-Slope Formula (6,16) , (14,14)

$(6, 16)$ , $(14, 14)$

Step 1

Find the slope of the line between $(6, 16)$ and $(14, 14)$ 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.

$m = \frac{14 - (16)}{14 - (6)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $16$ .

$m = \frac{14 - 16}{14 - (6)}$

Subtract $16$ from $14$ .

$m = \frac{- 2}{14 - (6)}$

Simplify the denominator.

Multiply $- 1$ by $6$ .

$m = \frac{- 2}{14 - 6}$

Subtract $6$ from $14$ .

$m = \frac{- 2}{8}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2$ and $8$ .

Factor $2$ out of $- 2$ .

$m = \frac{2 (- 1)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 2

Use the slope $- \frac{1}{4}$ and a given point $(6, 16)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (16) = - \frac{1}{4} \cdot (x - (6))$

Step 3

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

$y - 16 = - \frac{1}{4} \cdot (x - 6)$

Step 4

Solve for $y$ .

Simplify $- \frac{1}{4} \cdot (x - 6)$ .

Rewrite.

$y - 16 = 0 + 0 - \frac{1}{4} \cdot (x - 6)$

Simplify by adding zeros.

$y - 16 = - \frac{1}{4} \cdot (x - 6)$

Apply the distributive property.

$y - 16 = - \frac{1}{4} x - \frac{1}{4} \cdot - 6$

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

$y - 16 = - \frac{x}{4} - \frac{1}{4} \cdot - 6$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y - 16 = - \frac{x}{4} + \frac{- 1}{4} \cdot - 6$

Factor $2$ out of $4$ .

$y - 16 = - \frac{x}{4} + \frac{- 1}{2 (2)} \cdot - 6$

Factor $2$ out of $- 6$ .

$y - 16 = - \frac{x}{4} + \frac{- 1}{2 \cdot 2} \cdot (2 \cdot - 3)$

Cancel the common factor.

$y - 16 = - \frac{x}{4} + \frac{- 1}{2 \cdot 2} \cdot (2 \cdot - 3)$

Rewrite the expression.

$y - 16 = - \frac{x}{4} + \frac{- 1}{2} \cdot - 3$

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

$y - 16 = - \frac{x}{4} + \frac{- - 3}{2}$

Multiply $- 1$ by $- 3$ .

$y - 16 = - \frac{x}{4} + \frac{3}{2}$

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

Add $16$ to both sides of the equation.

$y = - \frac{x}{4} + \frac{3}{2} + 16$

To write $16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{4} + \frac{3}{2} + 16 \cdot \frac{2}{2}$

Combine $16$ and $\frac{2}{2}$ .

$y = - \frac{x}{4} + \frac{3}{2} + \frac{16 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{x}{4} + \frac{3 + 16 \cdot 2}{2}$

Simplify the numerator.

Multiply $16$ by $2$ .

$y = - \frac{x}{4} + \frac{3 + 32}{2}$

Add $3$ and $32$ .

$y = - \frac{x}{4} + \frac{35}{2}$

Step 5

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{35}{2}$

Step 6

Remove parentheses.

$y = - \frac{1}{4} x + \frac{35}{2}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{4} x + \frac{35}{2}$

Point-slope form:

$y - 16 = - \frac{1}{4} \cdot (x - 6)$

Step 8

Find the Equation Using Point-Slope Formula (6,16) , (14,14)

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