Finite Math Examples

$x + 8 y = 11$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $x + 8 y = 11$ .

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Step 2.1

Subtract $x$ from both sides of the equation.

$8 y = 11 - x$

Step 2.2

Divide each term in $8 y = 11 - x$ by $8$ and simplify.

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Step 2.2.1

Divide each term in $8 y = 11 - x$ by $8$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $8$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $y$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

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

Step 3

Find the slope when $y = \frac{11}{8} - \frac{x}{8}$ .

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Step 3.1

Rewrite in slope-intercept form.

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Step 3.1.1

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 3.1.2

Reorder $\frac{11}{8}$ and $- \frac{x}{8}$ .

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

Step 3.1.3

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

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Step 3.1.3.1

Reorder terms.

$y = - (\frac{1}{8} x) + \frac{11}{8}$

Step 3.1.3.2

Remove parentheses.

$y = - \frac{1}{8} x + \frac{11}{8}$

Step 3.2

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

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

Step 4

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 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 8$

Step 5.3

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

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Step 5.3.1

Multiply $8$ by $1$ .

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

Step 5.3.2

Multiply $- 1$ by $8$ .

$m_{perpendicular} = 8$

Step 5.3.3

Multiply $- 1$ by $- 8$ .

$m_{perpendicular} = 8$

Step 6

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

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Step 6.1

Use the slope $8$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 8 \cdot (x - (0))$

Step 6.2

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

$y + 0 = 8 \cdot (x + 0)$

Step 7

Solve for $y$ .

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Step 7.1

Add $y$ and $0$ .

$y = 8 \cdot (x + 0)$

Step 7.2

Add $x$ and $0$ .

$y = 8 x$

Step 8