Algebra Examples

$(1, - 8)$ and $(- 4, 4)$

Step 1

Find the slope of the line between $(1, - 8)$ and $(- 4, 4)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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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.

$m = \frac{4 - (- 8)}{- 4 - (1)}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $- 8$ .

$m = \frac{4 + 8}{- 4 - (1)}$

Step 1.4.1.2

Add $4$ and $8$ .

$m = \frac{12}{- 4 - (1)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $1$ .

$m = \frac{12}{- 4 - 1}$

Step 1.4.2.2

Subtract $1$ from $- 4$ .

$m = \frac{12}{- 5}$

Step 1.4.3

Move the negative in front of the fraction.

$m = - \frac{12}{5}$

Step 2

Use the slope $- \frac{12}{5}$ and a given point $(1, - 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 8) = - \frac{12}{5} \cdot (x - (1))$

Step 3

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

$y + 8 = - \frac{12}{5} \cdot (x - 1)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{12}{5} \cdot (x - 1)$ .

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Step 4.1.1

Rewrite.

$y + 8 = 0 + 0 - \frac{12}{5} \cdot (x - 1)$

Step 4.1.2

Simplify by adding zeros.

$y + 8 = - \frac{12}{5} \cdot (x - 1)$

Step 4.1.3

Apply the distributive property.

$y + 8 = - \frac{12}{5} x - \frac{12}{5} \cdot - 1$

Step 4.1.4

Combine $x$ and $\frac{12}{5}$ .

$y + 8 = - \frac{x \cdot 12}{5} - \frac{12}{5} \cdot - 1$

Step 4.1.5

Multiply $- \frac{12}{5} \cdot - 1$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

$y + 8 = - \frac{x \cdot 12}{5} + 1 (\frac{12}{5})$

Step 4.1.5.2

Multiply $\frac{12}{5}$ by $1$ .

$y + 8 = - \frac{x \cdot 12}{5} + \frac{12}{5}$

Step 4.1.6

Move $12$ to the left of $x$ .

$y + 8 = - \frac{12 x}{5} + \frac{12}{5}$

Step 4.2

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

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Step 4.2.1

Subtract $8$ from both sides of the equation.

$y = - \frac{12 x}{5} + \frac{12}{5} - 8$

Step 4.2.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{12 x}{5} + \frac{12}{5} - 8 \cdot \frac{5}{5}$

Step 4.2.3

Combine $- 8$ and $\frac{5}{5}$ .

$y = - \frac{12 x}{5} + \frac{12}{5} + \frac{- 8 \cdot 5}{5}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{12 x}{5} + \frac{12 - 8 \cdot 5}{5}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 8$ by $5$ .

$y = - \frac{12 x}{5} + \frac{12 - 40}{5}$

Step 4.2.5.2

Subtract $40$ from $12$ .

$y = - \frac{12 x}{5} + \frac{- 28}{5}$

Step 4.2.6

Move the negative in front of the fraction.

$y = - \frac{12 x}{5} - \frac{28}{5}$

Step 5

Reorder terms.

$y = - (\frac{12}{5} x) - \frac{28}{5}$

Step 6

Remove parentheses.

$y = - \frac{12}{5} x - \frac{28}{5}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{12}{5} x - \frac{28}{5}$

Point-slope form:

$y + 8 = - \frac{12}{5} \cdot (x - 1)$

Step 8