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

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

Step 1

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

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

Simplify.

Multiply the numerator and denominator of the complex fraction by $6$ .

Multiply $\frac{3 - - \frac{1}{3}}{\frac{5}{6} - \frac{1}{6}}$ by $\frac{6}{6}$ .

$m = \frac{6}{6} \cdot \frac{3 + \frac{1}{3}}{\frac{5}{6} - \frac{1}{6}}$

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $6$ by $3$ .

$m = \frac{18 + 6 (\frac{1}{3})}{5 - 1}$

Multiply $- - \frac{1}{3}$ .

Multiply $- 1$ by $- 1$ .

$m = \frac{18 + 6 (1 (\frac{1}{3}))}{5 - 1}$

Multiply $\frac{1}{3}$ by $1$ .

$m = \frac{18 + 6 (\frac{1}{3})}{5 - 1}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$m = \frac{18 + 3 (2) (\frac{1}{3})}{5 - 1}$

Cancel the common factor.

$m = \frac{18 + 3 \cdot (2 (\frac{1}{3}))}{5 - 1}$

Rewrite the expression.

$m = \frac{18 + 2}{5 - 1}$

Add $18$ and $2$ .

$m = \frac{20}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$m = \frac{20}{4}$

Divide $20$ by $4$ .

$m = 5$

Step 2

Use the slope $5$ and a given point $(\frac{1}{6}, - \frac{1}{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{1}{3}) = 5 \cdot (x - (\frac{1}{6}))$

Step 3

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

$y + \frac{1}{3} = 5 \cdot (x - \frac{1}{6})$

Step 4

Solve for $y$ .

Simplify $5 \cdot (x - \frac{1}{6})$ .

Rewrite.

$y + \frac{1}{3} = 0 + 0 + 5 \cdot (x - \frac{1}{6})$

Simplify by adding zeros.

$y + \frac{1}{3} = 5 \cdot (x - \frac{1}{6})$

Apply the distributive property.

$y + \frac{1}{3} = 5 x + 5 (- \frac{1}{6})$

Multiply $5 (- \frac{1}{6})$ .

Multiply $- 1$ by $5$ .

$y + \frac{1}{3} = 5 x - 5 (\frac{1}{6})$

Combine $- 5$ and $\frac{1}{6}$ .

$y + \frac{1}{3} = 5 x + \frac{- 5}{6}$

Move the negative in front of the fraction.

$y + \frac{1}{3} = 5 x - \frac{5}{6}$

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

Subtract $\frac{1}{3}$ from both sides of the equation.

$y = 5 x - \frac{5}{6} - \frac{1}{3}$

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 5 x - \frac{5}{6} - \frac{1}{3} \cdot \frac{2}{2}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$y = 5 x - \frac{5}{6} - \frac{2}{3 \cdot 2}$

Multiply $3$ by $2$ .

$y = 5 x - \frac{5}{6} - \frac{2}{6}$

Combine the numerators over the common denominator.

$y = 5 x + \frac{- 5 - 2}{6}$

Subtract $2$ from $- 5$ .

$y = 5 x + \frac{- 7}{6}$

Move the negative in front of the fraction.

$y = 5 x - \frac{7}{6}$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 5 x - \frac{7}{6}$

Point-slope form:

$y + \frac{1}{3} = 5 \cdot (x - \frac{1}{6})$

Step 6

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

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