Find the Equation Using Two Points (6,-6) , (8,8)

$(6, - 6)$ , $(8, 8)$

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.

$m = \frac{8 - (- 6)}{8 - (6)}$

Step 5

Finding the slope $m$ .

Cancel the common factor of $8 - (- 6)$ and $8 - (6)$ .

Rewrite $8$ as $- 1 (- 8)$ .

$m = \frac{8 - (- 6)}{- 1 \cdot - 8 - (6)}$

Factor $- 1$ out of $- 1 (- 8) - (6)$ .

$m = \frac{8 - (- 6)}{- 1 (- 8 + 6)}$

Reorder terms.

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

Factor $2$ out of $8$ .

$m = \frac{2 (4) - 6 \cdot - 1}{- 1 (- 8 + 6)}$

Factor $2$ out of $- 6 \cdot - 1$ .

$m = \frac{2 (4) + 2 (- 3 \cdot - 1)}{- 1 (- 8 + 6)}$

Factor $2$ out of $2 (4) + 2 (- 3 \cdot - 1)$ .

$m = \frac{2 (4 - 3 \cdot - 1)}{- 1 (- 8 + 6)}$

Cancel the common factors.

Factor $2$ out of $- 1 (- 8 + 6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $- 3$ by $- 1$ .

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

Add $4$ and $3$ .

$m = \frac{7}{- 1 (- 4 + 3)}$

Simplify the expression.

Add $- 4$ and $3$ .

$m = \frac{7}{- 1 \cdot - 1}$

Multiply $- 1$ by $- 1$ .

$m = \frac{7}{1}$

Divide $7$ by $1$ .

$m = 7$

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.

$y = (7) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (7) \cdot (6) + b$

Substitute the value of $y$ into the equation.

$- 6 = (7) \cdot (6) + b$

Find the value of $b$ .

Rewrite the equation as $(7) \cdot (6) + b = - 6$ .

$(7) \cdot (6) + b = - 6$

Multiply $7$ by $6$ .

$42 + b = - 6$

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

Subtract $42$ from both sides of the equation.

$b = - 6 - 42$

Subtract $42$ from $- 6$ .

$b = - 48$

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.

$y = 7 x - 48$

Step 8

Find the Equation Using Two Points (6,-6) , (8,8)

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