Precalculus Examples

(- 2, - 7)

$(- 2, - 7)$ ,

y = - 3 x

$y = - 3 x$

Step 1

Use the slope-intercept form to find the slope.

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Step 1.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 1.2

Using the slope-intercept form, the slope is

- 3

$- 3$ .

m = - 3

$m = - 3$

m = - 3

$m = - 3$

Step 2

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

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

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

Step 3

Simplify

- \frac{1}{- 3}

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

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Step 3.1

Move the negative in front of the fraction.

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

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

Step 3.2

Multiply

- - \frac{1}{3}

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

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Step 3.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.2.2

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

1

$1$ .

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

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

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

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

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

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

Step 4

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

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Step 4.1

Use the slope

\frac{1}{3}

$\frac{1}{3}$ and a given point

(- 2, - 7)

$(- 2, - 7)$ to substitute for

x_{1}

$x_{1}$ and

y_{1}

$y_{1}$ in the point-slope form

y - y_{1} = m (x - x_{1})

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

y - (- 7) = \frac{1}{3} \cdot (x - (- 2))

$y - (- 7) = \frac{1}{3} \cdot (x - (- 2))$

Step 4.2

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

y + 7 = \frac{1}{3} \cdot (x + 2)

$y + 7 = \frac{1}{3} \cdot (x + 2)$

y + 7 = \frac{1}{3} \cdot (x + 2)

$y + 7 = \frac{1}{3} \cdot (x + 2)$

Step 5

Write in

y = m x + b

$y = m x + b$ form.

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Step 5.1

Solve for

y

$y$ .

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Step 5.1.1

Simplify

\frac{1}{3} \cdot (x + 2)

$\frac{1}{3} \cdot (x + 2)$ .

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Step 5.1.1.1

Rewrite.

y + 7 = 0 + 0 + \frac{1}{3} \cdot (x + 2)

$y + 7 = 0 + 0 + \frac{1}{3} \cdot (x + 2)$

Step 5.1.1.2

Simplify by adding zeros.

y + 7 = \frac{1}{3} \cdot (x + 2)

$y + 7 = \frac{1}{3} \cdot (x + 2)$

Step 5.1.1.3

Apply the distributive property.

y + 7 = \frac{1}{3} x + \frac{1}{3} \cdot 2

$y + 7 = \frac{1}{3} x + \frac{1}{3} \cdot 2$

Step 5.1.1.4

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x

$x$ .

y + 7 = \frac{x}{3} + \frac{1}{3} \cdot 2

$y + 7 = \frac{x}{3} + \frac{1}{3} \cdot 2$

Step 5.1.1.5

Combine

\frac{1}{3}

$\frac{1}{3}$ and

2

$2$ .

y + 7 = \frac{x}{3} + \frac{2}{3}

$y + 7 = \frac{x}{3} + \frac{2}{3}$

y + 7 = \frac{x}{3} + \frac{2}{3}

$y + 7 = \frac{x}{3} + \frac{2}{3}$

Step 5.1.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 5.1.2.1

Subtract

7

$7$ from both sides of the equation.

y = \frac{x}{3} + \frac{2}{3} - 7

$y = \frac{x}{3} + \frac{2}{3} - 7$

Step 5.1.2.2

To write

- 7

$- 7$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

y = \frac{x}{3} + \frac{2}{3} - 7 \cdot \frac{3}{3}

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

Step 5.1.2.3

Combine

- 7

$- 7$ and

\frac{3}{3}

$\frac{3}{3}$ .

y = \frac{x}{3} + \frac{2}{3} + \frac{- 7 \cdot 3}{3}

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

Step 5.1.2.4

Combine the numerators over the common denominator.

y = \frac{x}{3} + \frac{2 - 7 \cdot 3}{3}

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

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply

- 7

$- 7$ by

3

$3$ .

y = \frac{x}{3} + \frac{2 - 21}{3}

$y = \frac{x}{3} + \frac{2 - 21}{3}$

Step 5.1.2.5.2

Subtract

21

$21$ from

2

$2$ .

y = \frac{x}{3} + \frac{- 19}{3}

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

y = \frac{x}{3} + \frac{- 19}{3}

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

Step 5.1.2.6

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 5.2

Reorder terms.

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

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

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

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

Step 6

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