Find Any Equation Perpendicular to the Line

$9 x = 2 y - 1$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $9 x = 2 y - 1$ .

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Step 2.1

Rewrite the equation as $2 y - 1 = 9 x$ .

$2 y - 1 = 9 x$

Step 2.2

Add $1$ to both sides of the equation.

$2 y = 9 x + 1$

Step 2.3

Divide each term in $2 y = 9 x + 1$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 y = 9 x + 1$ by $2$ .

$\frac{2 y}{2} = \frac{9 x}{2} + \frac{1}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{9 x}{2} + \frac{1}{2}$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{9 x}{2} + \frac{1}{2}$

Step 3

Find the slope when $y = \frac{9 x}{2} + \frac{1}{2}$ .

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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.

$y = \frac{9}{2} x + \frac{1}{2}$

Step 3.2

Using the slope-intercept form, the slope is $\frac{9}{2}$ .

$m = \frac{9}{2}$

Step 4

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

$m_{perpendicular} = - \frac{1}{\frac{9}{2}}$

Step 5

Simplify $- \frac{1}{\frac{9}{2}}$ to find the slope of the perpendicular line.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{2}{9}))$

Step 5.2

Multiply $\frac{2}{9}$ by $1$ .

$m_{perpendicular} = - \frac{2}{9}$

Step 6

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

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Step 6.1

Use the slope $- \frac{2}{9}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - \frac{2}{9} \cdot (x - (0))$

Step 6.2

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

$y + 0 = - \frac{2}{9} \cdot (x + 0)$

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 $0$ .

$y = - \frac{2}{9} \cdot (x + 0)$

Step 7.1.2

Simplify $- \frac{2}{9} \cdot (x + 0)$ .

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Step 7.1.2.1

Add $x$ and $0$ .

$y = - \frac{2}{9} \cdot x$

Step 7.1.2.2

Combine $x$ and $\frac{2}{9}$ .

$y = - \frac{x \cdot 2}{9}$

Step 7.1.2.3

Move $2$ to the left of $x$ .

$y = - \frac{2 x}{9}$

Step 7.2

Reorder terms.

$y = - (\frac{2}{9} x)$

Step 7.3

Remove parentheses.

$y = - \frac{2}{9} x$

Step 8

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Find Any Equation Perpendicular to the Line

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