Algebra Examples

Passing through

$(8, - 6)$ and perpendicular to the line whose equation is

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

Step 1

Find the slope when

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

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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

$y = m x + b$ , where

$m$ is the slope and

$b$ is the y-intercept.

$y = m x + b$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Combine

$\frac{1}{4}$ and

$x$ .

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

Step 1.1.3

Reorder terms.

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

Step 1.2

Using the slope-intercept form, the slope is

$\frac{1}{4}$ .

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

Step 2

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

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

Step 3

Simplify

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

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Multiply

$- (1 \cdot 4)$ .

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Step 3.2.1

Multiply

$4$ by

$1$ .

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

Step 3.2.2

Multiply

$- 1$ by

$4$ .

$m_{perpendicular} = - 4$

Step 4

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

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Step 4.1

Use the slope

$- 4$ and a given point

$(8, - 6)$ 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

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

$y - (- 6) = - 4 \cdot (x - (8))$

Step 4.2

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

$y + 6 = - 4 \cdot (x - 8)$

Step 5

Solve for

$y$ .

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Step 5.1

Simplify

$- 4 \cdot (x - 8)$ .

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Step 5.1.1

Rewrite.

$y + 6 = 0 + 0 - 4 \cdot (x - 8)$

Step 5.1.2

Simplify by adding zeros.

$y + 6 = - 4 \cdot (x - 8)$

Step 5.1.3

Apply the distributive property.

$y + 6 = - 4 x - 4 \cdot - 8$

Step 5.1.4

Multiply

$- 4$ by

$- 8$ .

$y + 6 = - 4 x + 32$

Step 5.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 5.2.1

Subtract

$6$ from both sides of the equation.

$y = - 4 x + 32 - 6$

Step 5.2.2

Subtract

$6$ from

$32$ .

$y = - 4 x + 26$

Step 6