Algebra Examples

What is an equation of the line that passes through the point

(- 2, 7)

$(- 2, 7)$ and is perpendicular to the line

x - 4 y = 24

$x - 4 y = 24$ ?

Step 1

Solve

x - 4 y = 24

$x - 4 y = 24$ .

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Step 1.1

Subtract

x

$x$ from both sides of the equation.

- 4 y = 24 - x

$- 4 y = 24 - x$

Step 1.2

Divide each term in

- 4 y = 24 - x

$- 4 y = 24 - x$ by

- 4

$- 4$ and simplify.

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Step 1.2.1

Divide each term in

- 4 y = 24 - x

$- 4 y = 24 - x$ by

- 4

$- 4$ .

\frac{- 4 y}{- 4} = \frac{24}{- 4} + \frac{- x}{- 4}

$\frac{- 4 y}{- 4} = \frac{24}{- 4} + \frac{- x}{- 4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 4

$- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{- 4 y}{- 4} = \frac{24}{- 4} + \frac{- x}{- 4}

$\frac{- 4 y}{- 4} = \frac{24}{- 4} + \frac{- x}{- 4}$

Step 1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{24}{- 4} + \frac{- x}{- 4}

$y = \frac{24}{- 4} + \frac{- x}{- 4}$

y = \frac{24}{- 4} + \frac{- x}{- 4}

$y = \frac{24}{- 4} + \frac{- x}{- 4}$

y = \frac{24}{- 4} + \frac{- x}{- 4}

$y = \frac{24}{- 4} + \frac{- x}{- 4}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide

24

$24$ by

- 4

$- 4$ .

y = - 6 + \frac{- x}{- 4}

$y = - 6 + \frac{- x}{- 4}$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$

Step 2

Find the slope when

y = - 6 + \frac{x}{4}

$y = - 6 + \frac{x}{4}$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

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 2.1.2

Reorder

- 6

$- 6$ and

\frac{x}{4}

$\frac{x}{4}$ .

y = \frac{x}{4} - 6

$y = \frac{x}{4} - 6$

Step 2.1.3

Reorder terms.

y = \frac{1}{4} x - 6

$y = \frac{1}{4} x - 6$

y = \frac{1}{4} x - 6

$y = \frac{1}{4} x - 6$

Step 2.2

Using the slope-intercept form, the slope is

\frac{1}{4}

$\frac{1}{4}$ .

m = \frac{1}{4}

$m = \frac{1}{4}$

m = \frac{1}{4}

$m = \frac{1}{4}$

Step 3

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}{4}}

$m_{perpendicular} = - \frac{1}{\frac{1}{4}}$

Step 4

Simplify

- \frac{1}{\frac{1}{4}}

$- \frac{1}{\frac{1}{4}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

m_{perpendicular} = - (1 \cdot 4)

$m_{perpendicular} = - (1 \cdot 4)$

Step 4.2

Multiply

- (1 \cdot 4)

$- (1 \cdot 4)$ .

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Step 4.2.1

Multiply

4

$4$ by

1

$1$ .

m_{perpendicular} = - 1 \cdot 4

$m_{perpendicular} = - 1 \cdot 4$

Step 4.2.2

Multiply

- 1

$- 1$ by

4

$4$ .

m_{perpendicular} = - 4

$m_{perpendicular} = - 4$

m_{perpendicular} = - 4

$m_{perpendicular} = - 4$

m_{perpendicular} = - 4

$m_{perpendicular} = - 4$

Step 5

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

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Step 5.1

Use the slope

- 4

$- 4$ 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) = - 4 \cdot (x - (- 2))

$y - (7) = - 4 \cdot (x - (- 2))$

Step 5.2

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

y - 7 = - 4 \cdot (x + 2)

$y - 7 = - 4 \cdot (x + 2)$

y - 7 = - 4 \cdot (x + 2)

$y - 7 = - 4 \cdot (x + 2)$

Step 6

Solve for

y

$y$ .

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Step 6.1

Simplify

- 4 \cdot (x + 2)

$- 4 \cdot (x + 2)$ .

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Step 6.1.1

Rewrite.

y - 7 = 0 + 0 - 4 \cdot (x + 2)

$y - 7 = 0 + 0 - 4 \cdot (x + 2)$

Step 6.1.2

Simplify by adding zeros.

y - 7 = - 4 \cdot (x + 2)

$y - 7 = - 4 \cdot (x + 2)$

Step 6.1.3

Apply the distributive property.

y - 7 = - 4 x - 4 \cdot 2

$y - 7 = - 4 x - 4 \cdot 2$

Step 6.1.4

Multiply

- 4

$- 4$ by

2

$2$ .

y - 7 = - 4 x - 8

$y - 7 = - 4 x - 8$

y - 7 = - 4 x - 8

$y - 7 = - 4 x - 8$

Step 6.2

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 6.2.1

Add

7

$7$ to both sides of the equation.

y = - 4 x - 8 + 7

$y = - 4 x - 8 + 7$

Step 6.2.2

Add

- 8

$- 8$ and

7

$7$ .

y = - 4 x - 1

$y = - 4 x - 1$

y = - 4 x - 1

$y = - 4 x - 1$

y = - 4 x - 1

$y = - 4 x - 1$

Step 7