Precalculus Examples

$2 x - 3 y = 5$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $2 x - 3 y = 5$ .

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Step 2.1

Subtract $2 x$ from both sides of the equation.

$- 3 y = 5 - 2 x$

Step 2.2

Divide each term in $- 3 y = 5 - 2 x$ by $- 3$ and simplify.

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Step 2.2.1

Divide each term in $- 3 y = 5 - 2 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{5}{- 3} + \frac{- 2 x}{- 3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{5}{- 3} + \frac{- 2 x}{- 3}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{- 3} + \frac{- 2 x}{- 3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{5}{3} + \frac{- 2 x}{- 3}$

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{5}{3} + \frac{2 x}{3}$

Step 3

Find the slope when $y = - \frac{5}{3} + \frac{2 x}{3}$ .

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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

Reorder $- \frac{5}{3}$ and $\frac{2 x}{3}$ .

$y = \frac{2 x}{3} - \frac{5}{3}$

Step 3.1.3

Reorder terms.

$y = \frac{2}{3} x - \frac{5}{3}$

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

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2

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 $x$ and $\frac{3}{2}$ .

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

Step 7.1.2.3

Move $3$ to the left of $x$ .

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

Step 7.2

Reorder terms.

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

Step 7.3

Remove parentheses.

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

Step 8