Algebra Examples

Through

(5, - 4)

$(5, - 4)$ ; perpendicular to

5 y = x - 10

$5 y = x - 10$

Step 1

Divide each term in

5 y = x - 10

$5 y = x - 10$ by

5

$5$ and simplify.

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Step 1.1

Divide each term in

5 y = x - 10

$5 y = x - 10$ by

5

$5$ .

\frac{5 y}{5} = \frac{x}{5} + \frac{- 10}{5}

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 10}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

5

$5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 10}{5}$

Step 1.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{x}{5} + \frac{- 10}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide

$- 10$ by

$5$ .

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

Step 2

Find the slope when

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

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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

$y = m x + b$ , where

$m$ is the slope and

$b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Reorder terms.

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

Step 2.2

Using the slope-intercept form, the slope is

$\frac{1}{5}$ .

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

Step 3

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

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

Step 4

Simplify

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply

$- (1 \cdot 5)$ .

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Step 4.2.1

Multiply

$5$ by

$1$ .

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

Step 4.2.2

Multiply

$- 1$ by

$5$ .

$m_{perpendicular} = - 5$

Step 5

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

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Step 5.1

Use the slope

$- 5$ and a given point

$(5, - 4)$ to substitute for

$x_{1}$ and

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

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

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

$y - (- 4) = - 5 \cdot (x - (5))$

Step 5.2

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

$y + 4 = - 5 \cdot (x - 5)$

Step 6

Solve for

$y$ .

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Step 6.1

Simplify

$- 5 \cdot (x - 5)$ .

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Step 6.1.1

Rewrite.

$y + 4 = 0 + 0 - 5 \cdot (x - 5)$

Step 6.1.2

Simplify by adding zeros.

$y + 4 = - 5 \cdot (x - 5)$

Step 6.1.3

Apply the distributive property.

$y + 4 = - 5 x - 5 \cdot - 5$

Step 6.1.4

Multiply

$- 5$ by

$- 5$ .

$y + 4 = - 5 x + 25$

Step 6.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 6.2.1

Subtract

$4$ from both sides of the equation.

$y = - 5 x + 25 - 4$

Step 6.2.2

Subtract

$4$ from

$25$ .

$y = - 5 x + 21$

Step 7