Precalculus Examples

$2 x - 3 y = 5$

Step 1

Rewrite in slope-intercept form.

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Step 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 1.2

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

$- 3 y = 5 - 2 x$

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

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

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Reorder terms.

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

Step 2

Using the slope-intercept form, the slope is $\frac{2}{3}$ .

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

Step 3

The equation of a perpendicular line to $y = \frac{2}{3} x - \frac{5}{3}$ must have a slope that is the negative reciprocal of the original slope.

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

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $\frac{3}{2}$ by $1$ .

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

Step 5