Find the Equation Using Two Points (-2,10) , (10,-14)

$(- 2, 10)$ , $(10, - 14)$

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.

$m = \frac{- 14 - (10)}{10 - (- 2)}$

Step 5

Finding the slope $m$ .

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 14 - (10)$ and $10 - (- 2)$ .

Rewrite $- 14$ as $- 1 (14)$ .

$m = \frac{- 1 \cdot 14 - (10)}{10 - (- 2)}$

Factor $- 1$ out of $- 1 (14) - (10)$ .

$m = \frac{- 1 (14 + 10)}{10 - (- 2)}$

Reorder terms.

$m = \frac{- 1 (14 + 10)}{10 - 2 \cdot - 1}$

Factor $2$ out of $- 1 (14 + 10)$ .

$m = \frac{2 (- 1 (7 + 5))}{10 - 2 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $10$ .

$m = \frac{2 (- 1 (7 + 5))}{2 (5) - 2 \cdot - 1}$

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

$m = \frac{2 (- 1 (7 + 5))}{2 (5) + 2 (1)}$

Factor $2$ out of $2 (5) + 2 (- - 1)$ .

$m = \frac{2 (- 1 (7 + 5))}{2 (5 + 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (7 + 5))}{2 (5 + 1)}$

Rewrite the expression.

$m = \frac{- 1 (7 + 5)}{5 + 1}$

Add $7$ and $5$ .

$m = \frac{- 1 \cdot 12}{5 + 1}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 \cdot 12}{5 + 1}$

Add $5$ and $1$ .

$m = \frac{- 1 \cdot 12}{6}$

Simplify the expression.

Multiply $- 1$ by $12$ .

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

Divide $- 12$ by $6$ .

$m = - 2$

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.

$y = (- 2) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- 2) \cdot (- 2) + b$

Substitute the value of $y$ into the equation.

$10 = (- 2) \cdot (- 2) + b$

Find the value of $b$ .

Rewrite the equation as $(- 2) \cdot (- 2) + b = 10$ .

$(- 2) \cdot (- 2) + b = 10$

Multiply $- 2$ by $- 2$ .

$4 + b = 10$

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

Subtract $4$ from both sides of the equation.

$b = 10 - 4$

Subtract $4$ from $10$ .

$b = 6$

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.

$y = - 2 x + 6$

Step 8

Find the Equation Using Two Points (-2,10) , (10,-14)

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