Algebra Examples

$(- 6, 5)$ , $(- 3, - 3)$

Step 1

Find the slope of the line between $(- 6, 5)$ and $(- 3, - 3)$ 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.

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

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 $5$ .

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

Step 1.4.1.2

Subtract $5$ from $- 3$ .

$m = \frac{- 8}{- 3 - (- 6)}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

Multiply $- 1$ by $- 6$ .

$m = \frac{- 8}{- 3 + 6}$

Step 1.4.2.2

Add $- 3$ and $6$ .

$m = \frac{- 8}{3}$

Step 1.4.3

Move the negative in front of the fraction.

$m = - \frac{8}{3}$

Step 2

Use the slope $- \frac{8}{3}$ and a given point $(- 6, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{8}{3} \cdot (x - (- 6))$

Step 3

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

$y - 5 = - \frac{8}{3} \cdot (x + 6)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{8}{3} \cdot (x + 6)$ .

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Step 4.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{8}{3} \cdot (x + 6)$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Apply the distributive property.

$y - 5 = - \frac{8}{3} x - \frac{8}{3} \cdot 6$

Step 4.1.2.2

Combine $x$ and $\frac{8}{3}$ .

$y - 5 = - \frac{x \cdot 8}{3} - \frac{8}{3} \cdot 6$

Step 4.1.2.3

Cancel the common factor of $3$ .

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Step 4.1.2.3.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$y - 5 = - \frac{x \cdot 8}{3} + \frac{- 8}{3} \cdot 6$

Step 4.1.2.3.2

Factor $3$ out of $6$ .

$y - 5 = - \frac{x \cdot 8}{3} + \frac{- 8}{3} \cdot (3 (2))$

Step 4.1.2.3.3

Cancel the common factor.

$y - 5 = - \frac{x \cdot 8}{3} + \frac{- 8}{3} \cdot (3 \cdot 2)$

Step 4.1.2.3.4

Rewrite the expression.

$y - 5 = - \frac{x \cdot 8}{3} - 8 \cdot 2$

Step 4.1.2.4

Multiply $- 8$ by $2$ .

$y - 5 = - \frac{x \cdot 8}{3} - 16$

Step 4.1.3

Move $8$ to the left of $x$ .

$y - 5 = - \frac{8 x}{3} - 16$

Step 4.2

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

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Step 4.2.1

Add $5$ to both sides of the equation.

$y = - \frac{8 x}{3} - 16 + 5$

Step 4.2.2

Add $- 16$ and $5$ .

$y = - \frac{8 x}{3} - 11$

Step 5

Reorder terms.

$y = - (\frac{8}{3} x) - 11$

Step 6

Remove parentheses.

$y = - \frac{8}{3} x - 11$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{8}{3} x - 11$

Point-slope form:

$y - 5 = - \frac{8}{3} \cdot (x + 6)$

Step 8