Algebra Examples

through: $(- 3, - 3)$ , perp. to $y = - \frac{3}{8} x - 2$

Step 1

Find the slope when $y = - \frac{3}{8} x - 2$ .

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Move $3$ to the left of $x$ .

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

Step 1.1.3

Write in $y = m x + b$ form.

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Step 1.1.3.1

Reorder terms.

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

Step 1.1.3.2

Remove parentheses.

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

Step 1.2

Using the slope-intercept form, the slope is $- \frac{3}{8}$ .

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

Step 2

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{- \frac{3}{8}}$

Step 3

Simplify $- \frac{1}{- \frac{3}{8}}$ to find the slope of the perpendicular line.

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Step 3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{8}}$

Step 3.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{8}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{8}{3})$

Step 3.3

Multiply $\frac{8}{3}$ by $1$ .

$m_{perpendicular} = \frac{8}{3}$

Step 3.4

Multiply $- - \frac{8}{3}$ .

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{8}{3})$

Step 3.4.2

Multiply $\frac{8}{3}$ by $1$ .

$m_{perpendicular} = \frac{8}{3}$

Step 4

Find the equation of the perpendicular line using the point-slope formula.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Write in $y = m x + b$ form.

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Step 5.1

Solve for $y$ .

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Step 5.1.1

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

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Step 5.1.1.1

Rewrite.

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

Step 5.1.1.2

Simplify by adding zeros.

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.1.4

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

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

Step 5.1.1.5

Cancel the common factor of $3$ .

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Step 5.1.1.5.1

Cancel the common factor.

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

Step 5.1.1.5.2

Rewrite the expression.

$y + 3 = \frac{8 x}{3} + 8$

Step 5.1.2

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

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Step 5.1.2.1

Subtract $3$ from both sides of the equation.

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

Step 5.1.2.2

Subtract $3$ from $8$ .

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

Step 5.2

Reorder terms.

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

Step 6