Finite Math Examples

$(1, 2)$ , $y = - \frac{1}{8} x + \frac{1}{3}$

Step 1

Find the slope when $y = - \frac{1}{8} x + \frac{1}{3}$ .

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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

Combine $x$ and $\frac{1}{8}$ .

$y = - \frac{x}{8} + \frac{1}{3}$

Step 1.1.3

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

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Step 1.1.3.1

Reorder terms.

$y = - (\frac{1}{8} x) + \frac{1}{3}$

Step 1.1.3.2

Remove parentheses.

$y = - \frac{1}{8} x + \frac{1}{3}$

Step 1.2

Using the slope-intercept form, the slope is $- \frac{1}{8}$ .

$m = - \frac{1}{8}$

Step 2

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

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

Step 3

Simplify $- \frac{1}{- \frac{1}{8}}$ 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 $- 1 (- 1)$ .

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

Step 3.1.2

Move the negative in front of the fraction.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 8$

Step 3.3

Multiply $- - (1 \cdot 8)$ .

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Step 3.3.1

Multiply $8$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 8)$

Step 3.3.2

Multiply $- 1$ by $8$ .

$m_{perpendicular} = 8$

Step 3.3.3

Multiply $- 1$ by $- 8$ .

$m_{perpendicular} = 8$

Step 4

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

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Step 4.1

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

$y - (2) = 8 \cdot (x - (1))$

Step 4.2

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

$y - 2 = 8 \cdot (x - 1)$

Step 5

Solve for $y$ .

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Step 5.1

Simplify $8 \cdot (x - 1)$ .

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Step 5.1.1

Rewrite.

$y - 2 = 0 + 0 + 8 \cdot (x - 1)$

Step 5.1.2

Simplify by adding zeros.

$y - 2 = 8 \cdot (x - 1)$

Step 5.1.3

Apply the distributive property.

$y - 2 = 8 x + 8 \cdot - 1$

Step 5.1.4

Multiply $8$ by $- 1$ .

$y - 2 = 8 x - 8$

Step 5.2

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

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Step 5.2.1

Add $2$ to both sides of the equation.

$y = 8 x - 8 + 2$

Step 5.2.2

Add $- 8$ and $2$ .

$y = 8 x - 6$

Step 6