Precalculus Examples

$(- 2, - 7)$ , $y = - 3 x$

Use the slope-intercept form to find the slope.

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The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $- 3$ .

$m = - 3$

The equation of a perpendicular line to $y = - 3 x$ must have a slope that is the negative reciprocal of the original slope.

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

Simplify $- \frac{1}{- 3}$ to find the slope of the perpendicular line.

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Move the negative in front of the fraction.

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

Simplify $- - \frac{1}{3}$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

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

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Find the value of $b$ using the formula for the equation of a line.

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Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (\frac{1}{3}) x + b$

Substitute the value of $x$ into the equation.

$y = (\frac{1}{3}) \cdot (- 2) + b$

Substitute the value of $y$ into the equation.

$(- 7) = (\frac{1}{3}) \cdot (- 2) + b$

Find the value of $b$ .

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Rewrite the equation as $(\frac{1}{3}) \cdot (- 2) + b = - 7$ .

$(\frac{1}{3}) \cdot (- 2) + b = - 7$

Simplify each term.

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

$(\frac{1}{3}) (- 2) + b = - 7$

Simplify $(\frac{1}{3}) (- 2)$ .

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Write $- 2$ as a fraction with denominator $1$ .

$\frac{1}{3} \cdot \frac{- 2}{1} + b = - 7$

Multiply $\frac{1}{3}$ and $\frac{- 2}{1}$ .

$\frac{- 2}{3} + b = - 7$

Move the negative in front of the fraction.

$- \frac{2}{3} + b = - 7$

Move all terms not containing $b$ to the right side of the equation.

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Add $\frac{2}{3}$ to both sides of the equation.

$b = \frac{2}{3} - 7$

Simplify the right side of the equation.

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To write $- \frac{- 7}{1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$b = \frac{2}{3} + \frac{- 7}{1} \cdot \frac{3}{3}$

Write each expression with a common denominator of $3$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$b = \frac{2}{3} + \frac{- 7 \cdot 3}{1 \cdot 3}$

Multiply $3$ by $1$ .

$b = \frac{2}{3} + \frac{- 7 \cdot 3}{3}$

Combine the numerators over the common denominator.

$b = \frac{2 - 7 \cdot 3}{3}$

Simplify the numerator.

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Multiply $- 7$ by $3$ .

$b = \frac{2 - 21}{3}$

Subtract $21$ from $2$ .

$b = \frac{- 19}{3}$

Move the negative in front of the fraction.

$b = - \frac{19}{3}$

Now that the values of $m$ (slope) and $b$ (y-intercept) are known, substitute them into $y = m x + b$ to find the equation of the line.

$y = \frac{1}{3} x - \frac{19}{3}$

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