Find the Equation Using Two Points (5,2) , (-5,7)

$(5, 2)$ , $(- 5, 7)$

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.

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

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $7$ .

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

Simplify the denominator.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $- 5$ .

$m = \frac{5}{- 10}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $5$ and $- 10$ .

Factor $5$ out of $5$ .

$m = \frac{5 (1)}{- 10}$

Cancel the common factors.

Factor $5$ out of $- 10$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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.

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

Substitute the value of $x$ into the equation.

$y = (- \frac{1}{2}) \cdot (5) + b$

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

Rewrite the equation as $- \frac{1}{2} \cdot 5 + b = 2$ .

$- \frac{1}{2} \cdot 5 + b = 2$

Simplify each term.

Multiply $- \frac{1}{2} \cdot 5$ .

Multiply $5$ by $- 1$ .

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

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{- 5}{2} + b = 2$

Move the negative in front of the fraction.

$- \frac{5}{2} + b = 2$

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

Add $\frac{5}{2}$ to both sides of the equation.

$b = 2 + \frac{5}{2}$

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

$b = 2 \cdot \frac{2}{2} + \frac{5}{2}$

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

$b = \frac{2 \cdot 2}{2} + \frac{5}{2}$

Combine the numerators over the common denominator.

$b = \frac{2 \cdot 2 + 5}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$b = \frac{4 + 5}{2}$

Add $4$ and $5$ .

$b = \frac{9}{2}$

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.

$y = - \frac{1}{2} x + \frac{9}{2}$

Step 8

Find the Equation Using Two Points (5,2) , (-5,7)

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