Find the Equation Using Two Points (1/6,-1/3) , (5/6,5)

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

Step 1

Use $y = m x + b$ to calculate the equation of the line, where $m$ represents the slope and $b$ represents the y-intercept.

To calculate the equation of the line, use the $y = m x + b$ format.

Step 2

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{(change in y)}{(change in x)}$

Step 3

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).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 4

Substitute in the values of $x$ and $y$ into the equation to find the slope.

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

Step 5

Finding the slope $m$ .

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

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $6$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $6$ by $5$ .

$m = \frac{30 + 6 (\frac{1}{3})}{5 - 1}$

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

Multiply $- 1$ by $- 1$ .

$m = \frac{30 + 6 (1 (\frac{1}{3}))}{5 - 1}$

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

$m = \frac{30 + 6 (\frac{1}{3})}{5 - 1}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$m = \frac{30 + 3 (2) (\frac{1}{3})}{5 - 1}$

Cancel the common factor.

$m = \frac{30 + 3 \cdot (2 (\frac{1}{3}))}{5 - 1}$

Rewrite the expression.

$m = \frac{30 + 2}{5 - 1}$

Add $30$ and $2$ .

$m = \frac{32}{5 - 1}$

Simplify the expression.

Subtract $1$ from $5$ .

$m = \frac{32}{4}$

Divide $32$ by $4$ .

$m = 8$

Step 6

Find the value of $b$ using the formula for the equation of a line.

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (8) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (8) \cdot (\frac{1}{6}) + b$

Substitute the value of $y$ into the equation.

$- \frac{1}{3} = (8) \cdot (\frac{1}{6}) + b$

Find the value of $b$ .

Rewrite the equation as $8 \cdot \frac{1}{6} + b = - \frac{1}{3}$ .

$8 \cdot \frac{1}{6} + b = - \frac{1}{3}$

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$2 (4) \cdot \frac{1}{6} + b = - \frac{1}{3}$

Factor $2$ out of $6$ .

$2 \cdot 4 \cdot \frac{1}{2 \cdot 3} + b = - \frac{1}{3}$

Cancel the common factor.

$2 \cdot 4 \cdot \frac{1}{2 \cdot 3} + b = - \frac{1}{3}$

Rewrite the expression.

$4 \cdot \frac{1}{3} + b = - \frac{1}{3}$

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} + b = - \frac{1}{3}$

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

Subtract $\frac{4}{3}$ from both sides of the equation.

$b = - \frac{1}{3} - \frac{4}{3}$

Combine the numerators over the common denominator.

$b = \frac{- 1 - 4}{3}$

Subtract $4$ from $- 1$ .

$b = \frac{- 5}{3}$

Move the negative in front of the fraction.

$b = - \frac{5}{3}$

Step 7

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = 8 x - \frac{5}{3}$

Step 8

Find the Equation Using Two Points (1/6,-1/3) , (5/6,5)

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