Algebra Examples

(- 3, 1)

$(- 3, 1)$ ; perpendicular to

y = - \frac{2}{5} x - 4

$y = - \frac{2}{5} x - 4$

Step 1

Find the slope when

y = - \frac{2}{5} x - 4

$y = - \frac{2}{5} x - 4$ .

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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

Simplify each term.

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Step 1.1.2.1.1

Combine

x

$x$ and

\frac{2}{5}

$\frac{2}{5}$ .

y = - \frac{x \cdot 2}{5} - 4

$y = - \frac{x \cdot 2}{5} - 4$

Step 1.1.2.1.2

Move

2

$2$ to the left of

x

$x$ .

y = - \frac{2 x}{5} - 4

$y = - \frac{2 x}{5} - 4$

y = - \frac{2 x}{5} - 4

$y = - \frac{2 x}{5} - 4$

y = - \frac{2 x}{5} - 4

$y = - \frac{2 x}{5} - 4$

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{2}{5} x) - 4

$y = - (\frac{2}{5} x) - 4$

Step 1.1.3.2

Remove parentheses.

y = - \frac{2}{5} x - 4

$y = - \frac{2}{5} x - 4$

y = - \frac{2}{5} x - 4

$y = - \frac{2}{5} x - 4$

y = - \frac{2}{5} x - 4

$y = - \frac{2}{5} x - 4$

Step 1.2

Using the slope-intercept form, the slope is

- \frac{2}{5}

$- \frac{2}{5}$ .

m = - \frac{2}{5}

$m = - \frac{2}{5}$

m = - \frac{2}{5}

$m = - \frac{2}{5}$

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{2}{5}}

$m_{perpendicular} = - \frac{1}{- \frac{2}{5}}$

Step 3

Simplify

- \frac{1}{- \frac{2}{5}}

$- \frac{1}{- \frac{2}{5}}$ 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{2}{5}}

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{2}{5}}$

Step 3.1.2

Move the negative in front of the fraction.

m_{perpendicular} = \frac{1}{\frac{2}{5}}

$m_{perpendicular} = \frac{1}{\frac{2}{5}}$

m_{perpendicular} = \frac{1}{\frac{2}{5}}

$m_{perpendicular} = \frac{1}{\frac{2}{5}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

m_{perpendicular} = 1 (\frac{5}{2})

$m_{perpendicular} = 1 (\frac{5}{2})$

Step 3.3

Multiply

\frac{5}{2}

$\frac{5}{2}$ by

1

$1$ .

m_{perpendicular} = \frac{5}{2}

$m_{perpendicular} = \frac{5}{2}$

Step 3.4

Multiply

- - \frac{5}{2}

$- - \frac{5}{2}$ .

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Step 3.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

m_{perpendicular} = 1 (\frac{5}{2})

$m_{perpendicular} = 1 (\frac{5}{2})$

Step 3.4.2

Multiply

\frac{5}{2}

$\frac{5}{2}$ by

1

$1$ .

m_{perpendicular} = \frac{5}{2}

$m_{perpendicular} = \frac{5}{2}$

m_{perpendicular} = \frac{5}{2}

$m_{perpendicular} = \frac{5}{2}$

m_{perpendicular} = \frac{5}{2}

$m_{perpendicular} = \frac{5}{2}$

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{5}{2}

$\frac{5}{2}$ and a given point

(- 3, 1)

$(- 3, 1)$ 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) = \frac{5}{2} \cdot (x - (- 3))

$y - (1) = \frac{5}{2} \cdot (x - (- 3))$

Step 4.2

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

y - 1 = \frac{5}{2} \cdot (x + 3)

$y - 1 = \frac{5}{2} \cdot (x + 3)$

y - 1 = \frac{5}{2} \cdot (x + 3)

$y - 1 = \frac{5}{2} \cdot (x + 3)$

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{5}{2} \cdot (x + 3)

$\frac{5}{2} \cdot (x + 3)$ .

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Step 5.1.1.1

Rewrite.

y - 1 = 0 + 0 + \frac{5}{2} \cdot (x + 3)

$y - 1 = 0 + 0 + \frac{5}{2} \cdot (x + 3)$

Step 5.1.1.2

Simplify by adding zeros.

y - 1 = \frac{5}{2} \cdot (x + 3)

$y - 1 = \frac{5}{2} \cdot (x + 3)$

Step 5.1.1.3

Apply the distributive property.

y - 1 = \frac{5}{2} x + \frac{5}{2} \cdot 3

$y - 1 = \frac{5}{2} x + \frac{5}{2} \cdot 3$

Step 5.1.1.4

Combine

\frac{5}{2}

$\frac{5}{2}$ and

x

$x$ .

y - 1 = \frac{5 x}{2} + \frac{5}{2} \cdot 3

$y - 1 = \frac{5 x}{2} + \frac{5}{2} \cdot 3$

Step 5.1.1.5

Multiply

\frac{5}{2} \cdot 3

$\frac{5}{2} \cdot 3$ .

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Step 5.1.1.5.1

Combine

\frac{5}{2}

$\frac{5}{2}$ and

3

$3$ .

y - 1 = \frac{5 x}{2} + \frac{5 \cdot 3}{2}

$y - 1 = \frac{5 x}{2} + \frac{5 \cdot 3}{2}$

Step 5.1.1.5.2

Multiply

5

$5$ by

3

$3$ .

y - 1 = \frac{5 x}{2} + \frac{15}{2}

$y - 1 = \frac{5 x}{2} + \frac{15}{2}$

y - 1 = \frac{5 x}{2} + \frac{15}{2}

$y - 1 = \frac{5 x}{2} + \frac{15}{2}$

y - 1 = \frac{5 x}{2} + \frac{15}{2}

$y - 1 = \frac{5 x}{2} + \frac{15}{2}$

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

Add

1

$1$ to both sides of the equation.

y = \frac{5 x}{2} + \frac{15}{2} + 1

$y = \frac{5 x}{2} + \frac{15}{2} + 1$

Step 5.1.2.2

Write

1

$1$ as a fraction with a common denominator.

y = \frac{5 x}{2} + \frac{15}{2} + \frac{2}{2}

$y = \frac{5 x}{2} + \frac{15}{2} + \frac{2}{2}$

Step 5.1.2.3

Combine the numerators over the common denominator.

y = \frac{5 x}{2} + \frac{15 + 2}{2}

$y = \frac{5 x}{2} + \frac{15 + 2}{2}$

Step 5.1.2.4

Add

15

$15$ and

2

$2$ .

y = \frac{5 x}{2} + \frac{17}{2}

$y = \frac{5 x}{2} + \frac{17}{2}$

y = \frac{5 x}{2} + \frac{17}{2}

$y = \frac{5 x}{2} + \frac{17}{2}$

y = \frac{5 x}{2} + \frac{17}{2}

$y = \frac{5 x}{2} + \frac{17}{2}$

Step 5.2

Reorder terms.

y = \frac{5}{2} x + \frac{17}{2}

$y = \frac{5}{2} x + \frac{17}{2}$

y = \frac{5}{2} x + \frac{17}{2}

$y = \frac{5}{2} x + \frac{17}{2}$

Step 6