Finite Math Examples

$2 x = - 3 y + 8$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve

$2 x = - 3 y + 8$ .

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Step 2.1

Rewrite the equation as

$- 3 y + 8 = 2 x$ .

$- 3 y + 8 = 2 x$

Step 2.2

Subtract

$8$ from both sides of the equation.

$- 3 y = 2 x - 8$

Step 2.3

Divide each term in

$- 3 y = 2 x - 8$ by

$- 3$ and simplify.

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Step 2.3.1

Divide each term in

$- 3 y = 2 x - 8$ by

$- 3$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of

$- 3$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide

$y$ by

$1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative in front of the fraction.

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

Step 2.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 3

Find the slope when

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

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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

Write in

$y = m x + b$ form.

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Step 3.1.2.1

Reorder terms.

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

Step 3.1.2.2

Remove parentheses.

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

Step 3.2

Using the slope-intercept form, the slope is

$- \frac{2}{3}$ .

$m = - \frac{2}{3}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{2}{3}}$

Step 5

Simplify

$- \frac{1}{- \frac{2}{3}}$ 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

$- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{3}}$

Step 5.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{2}{3}}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{2})$

Step 5.3

Multiply

$\frac{3}{2}$ by

$1$ .

$m_{perpendicular} = \frac{3}{2}$

Step 5.4

Multiply

$- - \frac{3}{2}$ .

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Step 5.4.1

Multiply

$- 1$ by

$- 1$ .

$m_{perpendicular} = 1 (\frac{3}{2})$

Step 5.4.2

Multiply

$\frac{3}{2}$ by

$1$ .

$m_{perpendicular} = \frac{3}{2}$

Step 6

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

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Step 6.1

Use the slope

$\frac{3}{2}$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{3}{2} \cdot (x - (0))$

Step 6.2

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

$y + 0 = \frac{3}{2} \cdot (x + 0)$

Step 7

Write in

$y = m x + b$ form.

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Step 7.1

Solve for

$y$ .

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Step 7.1.1

Add

$y$ and

$0$ .

$y = \frac{3}{2} \cdot (x + 0)$

Step 7.1.2

Simplify

$\frac{3}{2} \cdot (x + 0)$ .

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Step 7.1.2.1

Add

$x$ and

$0$ .

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

Step 7.1.2.2

Combine

$\frac{3}{2}$ and

$x$ .

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

Step 7.2

Reorder terms.

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

Step 8