Algebra Examples

Through

(8, - 5)

$(8, - 5)$ ; perpendicular to

7 y = x - 14

$7 y = x - 14$

Step 1

Divide each term in

7 y = x - 14

$7 y = x - 14$ by

7

$7$ and simplify.

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Step 1.1

Divide each term in

7 y = x - 14

$7 y = x - 14$ by

7

$7$ .

\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

7

$7$ .

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Step 1.2.1.1

Cancel the common factor.

\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}

$\frac{7 y}{7} = \frac{x}{7} + \frac{- 14}{7}$

Step 1.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{x}{7} + \frac{- 14}{7}

$y = \frac{x}{7} + \frac{- 14}{7}$

y = \frac{x}{7} + \frac{- 14}{7}

$y = \frac{x}{7} + \frac{- 14}{7}$

y = \frac{x}{7} + \frac{- 14}{7}

$y = \frac{x}{7} + \frac{- 14}{7}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide

- 14

$- 14$ by

7

$7$ .

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

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

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

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

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

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

Step 2

Find the slope when

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

$y = \frac{x}{7} - 2$ .

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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 terms.

y = \frac{1}{7} x - 2

$y = \frac{1}{7} x - 2$

y = \frac{1}{7} x - 2

$y = \frac{1}{7} x - 2$

Step 2.2

Using the slope-intercept form, the slope is

\frac{1}{7}

$\frac{1}{7}$ .

m = \frac{1}{7}

$m = \frac{1}{7}$

m = \frac{1}{7}

$m = \frac{1}{7}$

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}{7}}

$m_{perpendicular} = - \frac{1}{\frac{1}{7}}$

Step 4

Simplify

- \frac{1}{\frac{1}{7}}

$- \frac{1}{\frac{1}{7}}$ 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 7)

$m_{perpendicular} = - (1 \cdot 7)$

Step 4.2

Multiply

- (1 \cdot 7)

$- (1 \cdot 7)$ .

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Step 4.2.1

Multiply

7

$7$ by

1

$1$ .

m_{perpendicular} = - 1 \cdot 7

$m_{perpendicular} = - 1 \cdot 7$

Step 4.2.2

Multiply

- 1

$- 1$ by

7

$7$ .

m_{perpendicular} = - 7

$m_{perpendicular} = - 7$

m_{perpendicular} = - 7

$m_{perpendicular} = - 7$

m_{perpendicular} = - 7

$m_{perpendicular} = - 7$

Step 5

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

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Step 5.1

Use the slope

- 7

$- 7$ and a given point

(8, - 5)

$(8, - 5)$ 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 - (- 5) = - 7 \cdot (x - (8))

$y - (- 5) = - 7 \cdot (x - (8))$

Step 5.2

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

y + 5 = - 7 \cdot (x - 8)

$y + 5 = - 7 \cdot (x - 8)$

y + 5 = - 7 \cdot (x - 8)

$y + 5 = - 7 \cdot (x - 8)$

Step 6

Solve for

y

$y$ .

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Step 6.1

Simplify

- 7 \cdot (x - 8)

$- 7 \cdot (x - 8)$ .

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Step 6.1.1

Rewrite.

y + 5 = 0 + 0 - 7 \cdot (x - 8)

$y + 5 = 0 + 0 - 7 \cdot (x - 8)$

Step 6.1.2

Simplify by adding zeros.

y + 5 = - 7 \cdot (x - 8)

$y + 5 = - 7 \cdot (x - 8)$

Step 6.1.3

Apply the distributive property.

y + 5 = - 7 x - 7 \cdot - 8

$y + 5 = - 7 x - 7 \cdot - 8$

Step 6.1.4

Multiply

- 7

$- 7$ by

- 8

$- 8$ .

y + 5 = - 7 x + 56

$y + 5 = - 7 x + 56$

y + 5 = - 7 x + 56

$y + 5 = - 7 x + 56$

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

Subtract

5

$5$ from both sides of the equation.

y = - 7 x + 56 - 5

$y = - 7 x + 56 - 5$

Step 6.2.2

Subtract

5

$5$ from

56

$56$ .

y = - 7 x + 51

$y = - 7 x + 51$

y = - 7 x + 51

$y = - 7 x + 51$

y = - 7 x + 51

$y = - 7 x + 51$

Step 7