Find the Equation Using Two Points (-12,14) and (6,-1)

$(- 12, 14)$ and $(6, - 1)$

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.

$m = \frac{- 1 - (14)}{6 - (- 12)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $14$ .

$m = \frac{- 1 - 14}{6 - (- 12)}$

Subtract $14$ from $- 1$ .

$m = \frac{- 15}{6 - (- 12)}$

Simplify the denominator.

Multiply $- 1$ by $- 12$ .

$m = \frac{- 15}{6 + 12}$

Add $6$ and $12$ .

$m = \frac{- 15}{18}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 15$ and $18$ .

Factor $3$ out of $- 15$ .

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

Cancel the common factors.

Factor $3$ out of $18$ .

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

Cancel the common factor.

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

Rewrite the expression.

$m = \frac{- 5}{6}$

Move the negative in front of the fraction.

$m = - \frac{5}{6}$

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.

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

Substitute the value of $x$ into the equation.

$y = (- \frac{5}{6}) \cdot (- 12) + b$

Substitute the value of $y$ into the equation.

$14 = (- \frac{5}{6}) \cdot (- 12) + b$

Find the value of $b$ .

Rewrite the equation as $- \frac{5}{6} \cdot - 12 + b = 14$ .

$- \frac{5}{6} \cdot - 12 + b = 14$

Simplify each term.

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$\frac{- 5}{6} \cdot - 12 + b = 14$

Factor $6$ out of $- 12$ .

$\frac{- 5}{6} \cdot (6 (- 2)) + b = 14$

Cancel the common factor.

$\frac{- 5}{6} \cdot (6 \cdot - 2) + b = 14$

Rewrite the expression.

$- 5 \cdot - 2 + b = 14$

Multiply $- 5$ by $- 2$ .

$10 + b = 14$

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

Subtract $10$ from both sides of the equation.

$b = 14 - 10$

Subtract $10$ from $14$ .

$b = 4$

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.

$y = - \frac{5}{6} x + 4$

Step 8

Find the Equation Using Two Points (-12,14) and (6,-1)

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