Find the Equation Using Two Points (2,1) , (5,-8)

$(2, 1)$ , $(5, - 8)$

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.

$m = \frac{- 8 - (1)}{5 - (2)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $1$ .

$m = \frac{- 8 - 1}{5 - (2)}$

Subtract $1$ from $- 8$ .

$m = \frac{- 9}{5 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{- 9}{5 - 2}$

Subtract $2$ from $5$ .

$m = \frac{- 9}{3}$

Divide $- 9$ by $3$ .

$m = - 3$

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.

$y = (- 3) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- 3) \cdot (2) + b$

Substitute the value of $y$ into the equation.

$1 = (- 3) \cdot (2) + b$

Find the value of $b$ .

Rewrite the equation as $(- 3) \cdot (2) + b = 1$ .

$(- 3) \cdot (2) + b = 1$

Multiply $- 3$ by $2$ .

$- 6 + b = 1$

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

Add $6$ to both sides of the equation.

$b = 1 + 6$

Add $1$ and $6$ .

$b = 7$

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.

$y = - 3 x + 7$

Step 8

Find the Equation Using Two Points (2,1) , (5,-8)

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