Find the Equation Using Two Points (5,-2) , (-3,4)

$(5, - 2)$ , $(- 3, 4)$

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.

$m = \frac{4 - (- 2)}{- 3 - (5)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{4 + 2}{- 3 - (5)}$

Add $4$ and $2$ .

$m = \frac{6}{- 3 - (5)}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $- 3$ .

$m = \frac{6}{- 8}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $6$ and $- 8$ .

Factor $2$ out of $6$ .

$m = \frac{2 (3)}{- 8}$

Cancel the common factors.

Factor $2$ out of $- 8$ .

$m = \frac{2 \cdot 3}{2 \cdot - 4}$

Cancel the common factor.

$m = \frac{2 \cdot 3}{2 \cdot - 4}$

Rewrite the expression.

$m = \frac{3}{- 4}$

Move the negative in front of the fraction.

$m = - \frac{3}{4}$

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.

$y = (- \frac{3}{4}) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- \frac{3}{4}) \cdot (5) + b$

Substitute the value of $y$ into the equation.

$- 2 = (- \frac{3}{4}) \cdot (5) + b$

Find the value of $b$ .

Rewrite the equation as $- \frac{3}{4} \cdot 5 + b = - 2$ .

$- \frac{3}{4} \cdot 5 + b = - 2$

Simplify each term.

Multiply $- \frac{3}{4} \cdot 5$ .

Multiply $5$ by $- 1$ .

$- 5 (\frac{3}{4}) + b = - 2$

Combine $- 5$ and $\frac{3}{4}$ .

$\frac{- 5 \cdot 3}{4} + b = - 2$

Multiply $- 5$ by $3$ .

$\frac{- 15}{4} + b = - 2$

Move the negative in front of the fraction.

$- \frac{15}{4} + b = - 2$

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

Add $\frac{15}{4}$ to both sides of the equation.

$b = - 2 + \frac{15}{4}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$b = - 2 \cdot \frac{4}{4} + \frac{15}{4}$

Combine $- 2$ and $\frac{4}{4}$ .

$b = \frac{- 2 \cdot 4}{4} + \frac{15}{4}$

Combine the numerators over the common denominator.

$b = \frac{- 2 \cdot 4 + 15}{4}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

$b = \frac{- 8 + 15}{4}$

Add $- 8$ and $15$ .

$b = \frac{7}{4}$

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.

$y = - \frac{3}{4} x + \frac{7}{4}$

Step 8

Find the Equation Using Two Points (5,-2) , (-3,4)

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