Precalculus Examples

$(- 3, 4)$ , $(1, - 9)$

Step 1

Find the slope of the line between $(- 3, 4)$ and $(1, - 9)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Tap for more steps...

Step 1.1

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 1.2

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 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 9 - (4)}{1 - (- 3)}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Simplify the numerator.

Tap for more steps...

Step 1.4.1.1

Multiply $- 1$ by $4$ .

$m = \frac{- 9 - 4}{1 - (- 3)}$

Step 1.4.1.2

Subtract $4$ from $- 9$ .

$m = \frac{- 13}{1 - (- 3)}$

Step 1.4.2

Simplify the denominator.

Tap for more steps...

Step 1.4.2.1

Multiply $- 1$ by $- 3$ .

$m = \frac{- 13}{1 + 3}$

Step 1.4.2.2

Add $1$ and $3$ .

$m = \frac{- 13}{4}$

Step 1.4.3

Move the negative in front of the fraction.

$m = - \frac{13}{4}$

Step 2

Use the slope $- \frac{13}{4}$ and a given point $(- 3, 4)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{13}{4} \cdot (x - (- 3))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 4 = - \frac{13}{4} \cdot (x + 3)$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Simplify $- \frac{13}{4} \cdot (x + 3)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

$y - 4 = 0 + 0 - \frac{13}{4} \cdot (x + 3)$

Step 4.1.2

Simplify by adding zeros.

$y - 4 = - \frac{13}{4} \cdot (x + 3)$

Step 4.1.3

Apply the distributive property.

$y - 4 = - \frac{13}{4} x - \frac{13}{4} \cdot 3$

Step 4.1.4

Combine $x$ and $\frac{13}{4}$ .

$y - 4 = - \frac{x \cdot 13}{4} - \frac{13}{4} \cdot 3$

Step 4.1.5

Multiply $- \frac{13}{4} \cdot 3$ .

Tap for more steps...

Step 4.1.5.1

Multiply $3$ by $- 1$ .

$y - 4 = - \frac{x \cdot 13}{4} - 3 (\frac{13}{4})$

Step 4.1.5.2

Combine $- 3$ and $\frac{13}{4}$ .

$y - 4 = - \frac{x \cdot 13}{4} + \frac{- 3 \cdot 13}{4}$

Step 4.1.5.3

Multiply $- 3$ by $13$ .

$y - 4 = - \frac{x \cdot 13}{4} + \frac{- 39}{4}$

Step 4.1.6

Simplify each term.

Tap for more steps...

Step 4.1.6.1

Move $13$ to the left of $x$ .

$y - 4 = - \frac{13 \cdot x}{4} + \frac{- 39}{4}$

Step 4.1.6.2

Move the negative in front of the fraction.

$y - 4 = - \frac{13 x}{4} - \frac{39}{4}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Add $4$ to both sides of the equation.

$y = - \frac{13 x}{4} - \frac{39}{4} + 4$

Step 4.2.2

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

$y = - \frac{13 x}{4} - \frac{39}{4} + 4 \cdot \frac{4}{4}$

Step 4.2.3

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

$y = - \frac{13 x}{4} - \frac{39}{4} + \frac{4 \cdot 4}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{13 x}{4} + \frac{- 39 + 4 \cdot 4}{4}$

Step 4.2.5

Simplify the numerator.

Tap for more steps...

Step 4.2.5.1

Multiply $4$ by $4$ .

$y = - \frac{13 x}{4} + \frac{- 39 + 16}{4}$

Step 4.2.5.2

Add $- 39$ and $16$ .

$y = - \frac{13 x}{4} + \frac{- 23}{4}$

Step 4.2.6

Move the negative in front of the fraction.

$y = - \frac{13 x}{4} - \frac{23}{4}$

Step 5

Reorder terms.

$y = - (\frac{13}{4} x) - \frac{23}{4}$

Step 6

Remove parentheses.

$y = - \frac{13}{4} x - \frac{23}{4}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{13}{4} x - \frac{23}{4}$

Point-slope form:

$y - 4 = - \frac{13}{4} \cdot (x + 3)$

Step 8