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

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

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.

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

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

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

Add $5$ and $2$ .

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

Move the negative in front of the fraction.

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

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.

$y = (- \frac{1}{7}) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- \frac{1}{7}) \cdot (- 2) + b$

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

Rewrite the equation as $- \frac{1}{7} \cdot - 2 + b = 2$ .

$- \frac{1}{7} \cdot - 2 + b = 2$

Multiply $- \frac{1}{7} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

$2 (\frac{1}{7}) + b = 2$

Combine $2$ and $\frac{1}{7}$ .

$\frac{2}{7} + b = 2$

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

Subtract $\frac{2}{7}$ from both sides of the equation.

$b = 2 - \frac{2}{7}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$b = 2 \cdot \frac{7}{7} - \frac{2}{7}$

Combine $2$ and $\frac{7}{7}$ .

$b = \frac{2 \cdot 7}{7} - \frac{2}{7}$

Combine the numerators over the common denominator.

$b = \frac{2 \cdot 7 - 2}{7}$

Simplify the numerator.

Multiply $2$ by $7$ .

$b = \frac{14 - 2}{7}$

Subtract $2$ from $14$ .

$b = \frac{12}{7}$

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.

$y = - \frac{1}{7} x + \frac{12}{7}$

Step 8

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

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