Finite Math Examples

9 x = 4 y - 6

$9 x = 4 y - 6$

Step 1

Choose a point that the perpendicular line will pass through.

(0, 0)

$(0, 0)$

Step 2

Solve

9 x = 4 y - 6

$9 x = 4 y - 6$ .

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Step 2.1

Rewrite the equation as

4 y - 6 = 9 x

$4 y - 6 = 9 x$ .

4 y - 6 = 9 x

$4 y - 6 = 9 x$

Step 2.2

Add

6

$6$ to both sides of the equation.

4 y = 9 x + 6

$4 y = 9 x + 6$

Step 2.3

Divide each term in

4 y = 9 x + 6

$4 y = 9 x + 6$ by

4

$4$ and simplify.

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Step 2.3.1

Divide each term in

4 y = 9 x + 6

$4 y = 9 x + 6$ by

4

$4$ .

\frac{4 y}{4} = \frac{9 x}{4} + \frac{6}{4}

$\frac{4 y}{4} = \frac{9 x}{4} + \frac{6}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of

4

$4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{9 x}{4} + \frac{6}{4}$

Step 2.3.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{9 x}{4} + \frac{6}{4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of

$6$ and

$4$ .

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Step 2.3.3.1.1

Factor

$2$ out of

$6$ .

$y = \frac{9 x}{4} + \frac{2 (3)}{4}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor

$2$ out of

$4$ .

$y = \frac{9 x}{4} + \frac{2 \cdot 3}{2 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$y = \frac{9 x}{4} + \frac{2 \cdot 3}{2 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

$y = \frac{9 x}{4} + \frac{3}{2}$

Step 3

Find the slope when

$y = \frac{9 x}{4} + \frac{3}{2}$ .

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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 terms.

$y = \frac{9}{4} x + \frac{3}{2}$

Step 3.2

Using the slope-intercept form, the slope is

$\frac{9}{4}$ .

$m = \frac{9}{4}$

Step 4

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

$m_{perpendicular} = - \frac{1}{\frac{9}{4}}$

Step 5

Simplify

$- \frac{1}{\frac{9}{4}}$ to find the slope of the perpendicular line.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{4}{9}))$

Step 5.2

Multiply

$\frac{4}{9}$ by

$1$ .

$m_{perpendicular} = - \frac{4}{9}$

Step 6

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

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Step 6.1

Use the slope

$- \frac{4}{9}$ 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

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

$y - (0) = - \frac{4}{9} \cdot (x - (0))$

Step 6.2

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

$y + 0 = - \frac{4}{9} \cdot (x + 0)$

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

$0$ .

$y = - \frac{4}{9} \cdot (x + 0)$

Step 7.1.2

Simplify

$- \frac{4}{9} \cdot (x + 0)$ .

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Step 7.1.2.1

Add

$x$ and

$0$ .

$y = - \frac{4}{9} \cdot x$

Step 7.1.2.2

Combine

$x$ and

$\frac{4}{9}$ .

$y = - \frac{x \cdot 4}{9}$

Step 7.1.2.3

Move

$4$ to the left of

$x$ .

$y = - \frac{4 x}{9}$

Step 7.2

Reorder terms.

$y = - (\frac{4}{9} x)$

Step 7.3

Remove parentheses.

$y = - \frac{4}{9} x$

Step 8