Precalculus Examples

$y = 3 x - 21$ , $(7, 0)$

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 $y = 3 x - 21$ must have a slope that is the negative reciprocal of the original slope.

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

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.

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

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

$(0) = (- \frac{1}{3}) \cdot (7) + b$

Find the value of $b$ .

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Rewrite the equation as $(- \frac{1}{3}) \cdot (7) + b = 0$ .

$(- \frac{1}{3}) \cdot (7) + b = 0$

Simplify each term.

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Multiply $- \frac{1}{3}$ by $7$ to get $(- \frac{1}{3}) (7)$ .

$(- \frac{1}{3}) (7) + b = 0$

Simplify $(- \frac{1}{3}) (7)$ .

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Multiply $7$ by $- 1$ to get $- 7$ .

$- 7 (\frac{1}{3}) + b = 0$

Write $- 7$ as a fraction with denominator $1$ .

$\frac{- 7}{1} \cdot \frac{1}{3} + b = 0$

Multiply $\frac{- 7}{1}$ and $\frac{1}{3}$ to get $\frac{- 7}{3}$ .

$\frac{- 7}{3} + b = 0$

Move the negative in front of the fraction.

$- \frac{7}{3} + b = 0$

Add $\frac{7}{3}$ to both sides of the equation.

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

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.

$y = - \frac{1}{3} x + \frac{7}{3}$

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