Algebra Examples

A line is perpendicular to $y = - 4 x - 2$ and intersects the point $(0, 9)$ What is the equation of this perpendicular line?

Step 1

Write the problem as a mathematical expression.

$y = - 4 x - 2$ , $(0, 9)$

Step 2

Use the slope-intercept form to find the slope.

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Step 2.1

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 2.2

Using the slope-intercept form, the slope is $- 4$ .

$m = - 4$

Step 3

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

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

Step 4

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

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Multiply $- - \frac{1}{4}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{4})$

Step 4.2.2

Multiply $\frac{1}{4}$ by $1$ .

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

Step 5

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

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Step 5.1

Use the slope $\frac{1}{4}$ and a given point $(0, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = \frac{1}{4} \cdot (x - (0))$

Step 5.2

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

$y - 9 = \frac{1}{4} \cdot (x + 0)$

Step 6

Write in $y = m x + b$ form.

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Step 6.1

Solve for $y$ .

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Step 6.1.1

Simplify $\frac{1}{4} \cdot (x + 0)$ .

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Step 6.1.1.1

Add $x$ and $0$ .

$y - 9 = \frac{1}{4} \cdot x$

Step 6.1.1.2

Combine $\frac{1}{4}$ and $x$ .

$y - 9 = \frac{x}{4}$

Step 6.1.2

Add $9$ to both sides of the equation.

$y = \frac{x}{4} + 9$

Step 6.2

Reorder terms.

$y = \frac{1}{4} x + 9$

Step 7