Precalculus Examples

$2 x + y = 2$ , $(- 3, 0)$

Step 1

Subtract $2 x$ from both sides of the equation.

$y = 2 - 2 x$

Step 2

Find the slope when $y = 2 - 2 x$ .

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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

Reorder $2$ and $- 2 x$ .

$y = - 2 x + 2$

Step 2.2

Using the slope-intercept form, the slope is $- 2$ .

$m = - 2$

Step 3

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

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

Step 4

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

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Multiply $- - \frac{1}{2}$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 5

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

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Step 5.1

Use the slope $\frac{1}{2}$ and a given point $(- 3, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{1}{2} \cdot (x - (- 3))$

Step 5.2

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

$y + 0 = \frac{1}{2} \cdot (x + 3)$

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

Add $y$ and $0$ .

$y = \frac{1}{2} \cdot (x + 3)$

Step 6.1.2

Simplify $\frac{1}{2} \cdot (x + 3)$ .

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Step 6.1.2.1

Apply the distributive property.

$y = \frac{1}{2} x + \frac{1}{2} \cdot 3$

Step 6.1.2.2

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

$y = \frac{x}{2} + \frac{1}{2} \cdot 3$

Step 6.1.2.3

Combine $\frac{1}{2}$ and $3$ .

$y = \frac{x}{2} + \frac{3}{2}$

Step 6.2

Reorder terms.

$y = \frac{1}{2} x + \frac{3}{2}$

Step 7