Algebra Examples

Find the equation of the line through $(2, - 9)$ which is perpendicular to the line $y = - \frac{x}{2} - 4$

Step 1

Find the slope when $y = - \frac{x}{2} - 4$ .

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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

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

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Step 1.1.2.1

Reorder terms.

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

Step 1.1.2.2

Remove parentheses.

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

Step 1.2

Using the slope-intercept form, the slope is $- \frac{1}{2}$ .

$m = - \frac{1}{2}$

Step 2

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

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

Step 3

Simplify $- \frac{1}{- \frac{1}{2}}$ to find the slope of the perpendicular line.

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Step 3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.1.2

Move the negative in front of the fraction.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 2$

Step 3.3

Multiply $- - (1 \cdot 2)$ .

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Step 3.3.1

Multiply $2$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 2)$

Step 3.3.2

Multiply $- 1$ by $2$ .

$m_{perpendicular} = 2$

Step 3.3.3

Multiply $- 1$ by $- 2$ .

$m_{perpendicular} = 2$

Step 4

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

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Step 4.1

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

$y - (- 9) = 2 \cdot (x - (2))$

Step 4.2

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

$y + 9 = 2 \cdot (x - 2)$

Step 5

Solve for $y$ .

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Step 5.1

Simplify $2 \cdot (x - 2)$ .

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Step 5.1.1

Rewrite.

$y + 9 = 0 + 0 + 2 \cdot (x - 2)$

Step 5.1.2

Simplify by adding zeros.

$y + 9 = 2 \cdot (x - 2)$

Step 5.1.3

Apply the distributive property.

$y + 9 = 2 x + 2 \cdot - 2$

Step 5.1.4

Multiply $2$ by $- 2$ .

$y + 9 = 2 x - 4$

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 $9$ from both sides of the equation.

$y = 2 x - 4 - 9$

Step 5.2.2

Subtract $9$ from $- 4$ .

$y = 2 x - 13$

Step 6