Find the Equation Using Two Points (-4,2) , (2,6)

$(- 4, 2)$ , $(2, 6)$

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.

$m = \frac{6 - (2)}{2 - (- 4)}$

Step 5

Finding the slope $m$ .

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6 - (2)$ and $2 - (- 4)$ .

Rewrite $6$ as $- 1 (- 6)$ .

$m = \frac{- 1 \cdot - 6 - (2)}{2 - (- 4)}$

Factor $- 1$ out of $- 1 (- 6) - (2)$ .

$m = \frac{- 1 (- 6 + 2)}{2 - (- 4)}$

Reorder terms.

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

Factor $2$ out of $- 1 (- 6 + 2)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 4 \cdot - 1$ .

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

Factor $2$ out of $2 (1) + 2 (- 2 \cdot - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 3$ and $1$ .

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

Simplify the denominator.

Multiply $- 2$ by $- 1$ .

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

Add $1$ and $2$ .

$m = \frac{- 1 \cdot - 2}{3}$

Multiply $- 1$ by $- 2$ .

$m = \frac{2}{3}$

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.

$y = (\frac{2}{3}) x + b$

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

Rewrite the equation as $\frac{2}{3} \cdot - 4 + b = 2$ .

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

Simplify each term.

Multiply $\frac{2}{3} \cdot - 4$ .

Combine $\frac{2}{3}$ and $- 4$ .

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

Multiply $2$ by $- 4$ .

$\frac{- 8}{3} + b = 2$

Move the negative in front of the fraction.

$- \frac{8}{3} + b = 2$

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

Add $\frac{8}{3}$ to both sides of the equation.

$b = 2 + \frac{8}{3}$

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

$b = 2 \cdot \frac{3}{3} + \frac{8}{3}$

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

$b = \frac{2 \cdot 3}{3} + \frac{8}{3}$

Combine the numerators over the common denominator.

$b = \frac{2 \cdot 3 + 8}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$b = \frac{6 + 8}{3}$

Add $6$ and $8$ .

$b = \frac{14}{3}$

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.

$y = \frac{2}{3} x + \frac{14}{3}$

Step 8

Find the Equation Using Two Points (-4,2) , (2,6)

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