Algebra Examples

A line is perpendicular to $y = \frac{1}{2} x - 9$ and intersects the point $(3, 2)$ What is the equation of this perpendicular line?

Step 1

Write the problem as a mathematical expression.

$y = \frac{1}{2} x - 9$ , $(3, 2)$

Step 2

Find the slope when $y = \frac{1}{2} x - 9$ .

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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

Simplify the right side.

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Step 2.1.2.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.1.3

Reorder terms.

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $- (1 \cdot 2)$ .

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Step 4.2.1

Multiply $2$ by $1$ .

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

Step 4.2.2

Multiply $- 1$ by $2$ .

$m_{perpendicular} = - 2$

Step 5

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

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Step 5.1

Use the slope $- 2$ and a given point $(3, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - 2 \cdot (x - (3))$

Step 5.2

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

$y - 2 = - 2 \cdot (x - 3)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- 2 \cdot (x - 3)$ .

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Step 6.1.1

Rewrite.

$y - 2 = 0 + 0 - 2 \cdot (x - 3)$

Step 6.1.2

Simplify by adding zeros.

$y - 2 = - 2 \cdot (x - 3)$

Step 6.1.3

Apply the distributive property.

$y - 2 = - 2 x - 2 \cdot - 3$

Step 6.1.4

Multiply $- 2$ by $- 3$ .

$y - 2 = - 2 x + 6$

Step 6.2

Move all terms not containing $y$ to the right side of the equation.

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Step 6.2.1

Add $2$ to both sides of the equation.

$y = - 2 x + 6 + 2$

Step 6.2.2

Add $6$ and $2$ .

$y = - 2 x + 8$

Step 7