Algebra Examples

A line is perpendicular to

y = - \frac{1}{5} x + 1

$y = - \frac{1}{5} x + 1$ and intersects the point

(- 5, 1)

$(- 5, 1)$ What is the equation of this perpendicular line?

Step 1

Find the slope when

y = - \frac{1}{5} x + 1

$y = - \frac{1}{5} x + 1$ .

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

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 1.1.2

Simplify the right side.

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Step 1.1.2.1

Combine

x

$x$ and

\frac{1}{5}

$\frac{1}{5}$ .

y = - \frac{x}{5} + 1

$y = - \frac{x}{5} + 1$

y = - \frac{x}{5} + 1

$y = - \frac{x}{5} + 1$

Step 1.1.3

Write in

y = m x + b

$y = m x + b$ form.

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Step 1.1.3.1

Reorder terms.

y = - (\frac{1}{5} x) + 1

$y = - (\frac{1}{5} x) + 1$

Step 1.1.3.2

Remove parentheses.

y = - \frac{1}{5} x + 1

$y = - \frac{1}{5} x + 1$

y = - \frac{1}{5} x + 1

$y = - \frac{1}{5} x + 1$

y = - \frac{1}{5} x + 1

$y = - \frac{1}{5} x + 1$

Step 1.2

Using the slope-intercept form, the slope is

- \frac{1}{5}

$- \frac{1}{5}$ .

m = - \frac{1}{5}

$m = - \frac{1}{5}$

m = - \frac{1}{5}

$m = - \frac{1}{5}$

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}{- \frac{1}{5}}

$m_{perpendicular} = - \frac{1}{- \frac{1}{5}}$

Step 3

Simplify

- \frac{1}{- \frac{1}{5}}

$- \frac{1}{- \frac{1}{5}}$ to find the slope of the perpendicular line.

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Step 3.1

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

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Step 3.1.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{1}{5}}

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{1}{5}}$

Step 3.1.2

Move the negative in front of the fraction.

m_{perpendicular} = \frac{1}{\frac{1}{5}}

$m_{perpendicular} = \frac{1}{\frac{1}{5}}$

m_{perpendicular} = \frac{1}{\frac{1}{5}}

$m_{perpendicular} = \frac{1}{\frac{1}{5}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

m_{perpendicular} = 1 \cdot 5

$m_{perpendicular} = 1 \cdot 5$

Step 3.3

Multiply

- - (1 \cdot 5)

$- - (1 \cdot 5)$ .

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Step 3.3.1

Multiply

5

$5$ by

1

$1$ .

m_{perpendicular} = - (- 1 \cdot 5)

$m_{perpendicular} = - (- 1 \cdot 5)$

Step 3.3.2

Multiply

- 1

$- 1$ by

5

$5$ .

m_{perpendicular} = 5

$m_{perpendicular} = 5$

Step 3.3.3

Multiply

- 1

$- 1$ by

- 5

$- 5$ .

m_{perpendicular} = 5

$m_{perpendicular} = 5$

m_{perpendicular} = 5

$m_{perpendicular} = 5$

m_{perpendicular} = 5

$m_{perpendicular} = 5$

Step 4

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

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Step 4.1

Use the slope

5

$5$ and a given point

(- 5, 1)

$(- 5, 1)$ 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 - (1) = 5 \cdot (x - (- 5))

$y - (1) = 5 \cdot (x - (- 5))$

Step 4.2

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

y - 1 = 5 \cdot (x + 5)

$y - 1 = 5 \cdot (x + 5)$

y - 1 = 5 \cdot (x + 5)

$y - 1 = 5 \cdot (x + 5)$

Step 5

Solve for

y

$y$ .

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Step 5.1

Simplify

5 \cdot (x + 5)

$5 \cdot (x + 5)$ .

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Step 5.1.1

Rewrite.

y - 1 = 0 + 0 + 5 \cdot (x + 5)

$y - 1 = 0 + 0 + 5 \cdot (x + 5)$

Step 5.1.2

Simplify by adding zeros.

y - 1 = 5 \cdot (x + 5)

$y - 1 = 5 \cdot (x + 5)$

Step 5.1.3

Apply the distributive property.

y - 1 = 5 x + 5 \cdot 5

$y - 1 = 5 x + 5 \cdot 5$

Step 5.1.4

Multiply

5

$5$ by

5

$5$ .

y - 1 = 5 x + 25

$y - 1 = 5 x + 25$

y - 1 = 5 x + 25

$y - 1 = 5 x + 25$

Step 5.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 5.2.1

Add

1

$1$ to both sides of the equation.

y = 5 x + 25 + 1

$y = 5 x + 25 + 1$

Step 5.2.2

Add

25

$25$ and

1

$1$ .

y = 5 x + 26

$y = 5 x + 26$

y = 5 x + 26

$y = 5 x + 26$

y = 5 x + 26

$y = 5 x + 26$

Step 6