Finite Math Examples

$y - (- 12) = - \frac{1}{2} \cdot (x - (- 6))$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $y - (- 12) = - \frac{1}{2} \cdot (x - (- 6))$ .

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Step 2.1

Multiply $- 1$ by $- 12$ .

$y + 12 = - \frac{1}{2} \cdot (x - (- 6))$

Step 2.2

Simplify $- \frac{1}{2} \cdot (x - (- 6))$ .

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Step 2.2.1

Multiply $- 1$ by $- 6$ .

$y + 12 = - \frac{1}{2} \cdot (x + 6)$

Step 2.2.2

Apply the distributive property.

$y + 12 = - \frac{1}{2} x - \frac{1}{2} \cdot 6$

Step 2.2.3

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

$y + 12 = - \frac{x}{2} - \frac{1}{2} \cdot 6$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y + 12 = - \frac{x}{2} + \frac{- 1}{2} \cdot 6$

Step 2.2.4.2

Factor $2$ out of $6$ .

$y + 12 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 (3))$

Step 2.2.4.3

Cancel the common factor.

$y + 12 = - \frac{x}{2} + \frac{- 1}{2} \cdot (2 \cdot 3)$

Step 2.2.4.4

Rewrite the expression.

$y + 12 = - \frac{x}{2} - 1 \cdot 3$

Step 2.2.5

Multiply $- 1$ by $3$ .

$y + 12 = - \frac{x}{2} - 3$

Step 2.3

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

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Step 2.3.1

Subtract $12$ from both sides of the equation.

$y = - \frac{x}{2} - 3 - 12$

Step 2.3.2

Subtract $12$ from $- 3$ .

$y = - \frac{x}{2} - 15$

Step 3

Find the slope when $y = - \frac{x}{2} - 15$ .

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Step 3.1

Rewrite in slope-intercept form.

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Step 3.1.1

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 3.1.2

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

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Step 3.1.2.1

Reorder terms.

$y = - (\frac{1}{2} x) - 15$

Step 3.1.2.2

Remove parentheses.

$y = - \frac{1}{2} x - 15$

Step 3.2

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

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

Step 4

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 5

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

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 2$

Step 5.3

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

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Step 5.3.1

Multiply $2$ by $1$ .

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

Step 5.3.2

Multiply $- 1$ by $2$ .

$m_{perpendicular} = 2$

Step 5.3.3

Multiply $- 1$ by $- 2$ .

$m_{perpendicular} = 2$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Solve for $y$ .

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Step 7.1

Add $y$ and $0$ .

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

Step 7.2

Add $x$ and $0$ .

$y = 2 x$

Step 8