Algebra Examples

through:

$(- 1, - 4)$ , perp. to

$y = 2 x + 5$

Step 1

Use the slope-intercept form to find the slope.

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Step 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 1.2

Using the slope-intercept form, the slope is

$2$ .

$m = 2$

Step 2

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

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

Step 3

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

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Step 3.1

Use the slope

$- \frac{1}{2}$ and a given point

$(- 1, - 4)$ 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

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

$y - (- 4) = - \frac{1}{2} \cdot (x - (- 1))$

Step 3.2

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

$y + 4 = - \frac{1}{2} \cdot (x + 1)$

Step 4

Write in

$y = m x + b$ form.

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Step 4.1

Solve for

$y$ .

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Step 4.1.1

Simplify

$- \frac{1}{2} \cdot (x + 1)$ .

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Step 4.1.1.1

Rewrite.

$y + 4 = 0 + 0 - \frac{1}{2} \cdot (x + 1)$

Step 4.1.1.2

Simplify by adding zeros.

$y + 4 = - \frac{1}{2} \cdot (x + 1)$

Step 4.1.1.3

Apply the distributive property.

$y + 4 = - \frac{1}{2} x - \frac{1}{2} \cdot 1$

Step 4.1.1.4

Combine

$x$ and

$\frac{1}{2}$ .

$y + 4 = - \frac{x}{2} - \frac{1}{2} \cdot 1$

Step 4.1.1.5

Multiply

$- 1$ by

$1$ .

$y + 4 = - \frac{x}{2} - \frac{1}{2}$

Step 4.1.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 4.1.2.1

Subtract

$4$ from both sides of the equation.

$y = - \frac{x}{2} - \frac{1}{2} - 4$

Step 4.1.2.2

To write

$- 4$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y = - \frac{x}{2} - \frac{1}{2} - 4 \cdot \frac{2}{2}$

Step 4.1.2.3

Combine

$- 4$ and

$\frac{2}{2}$ .

$y = - \frac{x}{2} - \frac{1}{2} + \frac{- 4 \cdot 2}{2}$

Step 4.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{- 1 - 4 \cdot 2}{2}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Multiply

$- 4$ by

$2$ .

$y = - \frac{x}{2} + \frac{- 1 - 8}{2}$

Step 4.1.2.5.2

Subtract

$8$ from

$- 1$ .

$y = - \frac{x}{2} + \frac{- 9}{2}$

Step 4.1.2.6

Move the negative in front of the fraction.

$y = - \frac{x}{2} - \frac{9}{2}$

Step 4.2

Reorder terms.

$y = - (\frac{1}{2} x) - \frac{9}{2}$

Step 4.3

Remove parentheses.

$y = - \frac{1}{2} x - \frac{9}{2}$

Step 5