Finite Math Examples

$12 x + 4 y = 16$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $12 x + 4 y = 16$ .

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Step 2.1

Subtract $12 x$ from both sides of the equation.

$4 y = 16 - 12 x$

Step 2.2

Divide each term in $4 y = 16 - 12 x$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 y = 16 - 12 x$ by $4$ .

$\frac{4 y}{4} = \frac{16}{4} + \frac{- 12 x}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{16}{4} + \frac{- 12 x}{4}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{16}{4} + \frac{- 12 x}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $16$ by $4$ .

$y = 4 + \frac{- 12 x}{4}$

Step 2.2.3.1.2

Cancel the common factor of $- 12$ and $4$ .

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Step 2.2.3.1.2.1

Factor $4$ out of $- 12 x$ .

$y = 4 + \frac{4 (- 3 x)}{4}$

Step 2.2.3.1.2.2

Cancel the common factors.

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Step 2.2.3.1.2.2.1

Factor $4$ out of $4$ .

$y = 4 + \frac{4 (- 3 x)}{4 (1)}$

Step 2.2.3.1.2.2.2

Cancel the common factor.

$y = 4 + \frac{4 (- 3 x)}{4 \cdot 1}$

Step 2.2.3.1.2.2.3

Rewrite the expression.

$y = 4 + \frac{- 3 x}{1}$

Step 2.2.3.1.2.2.4

Divide $- 3 x$ by $1$ .

$y = 4 - 3 x$

Step 3

Find the slope when $y = 4 - 3 x$ .

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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

Reorder $4$ and $- 3 x$ .

$y = - 3 x + 4$

Step 3.2

Using the slope-intercept form, the slope is $- 3$ .

$m = - 3$

Step 4

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

$m_{perpendicular} = - \frac{1}{- 3}$

Step 5

Simplify $- \frac{1}{- 3}$ to find the slope of the perpendicular line.

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Step 5.1

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{3}$

Step 5.2

Multiply $- - \frac{1}{3}$ .

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{3})$

Step 5.2.2

Multiply $\frac{1}{3}$ by $1$ .

$m_{perpendicular} = \frac{1}{3}$

Step 6

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

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Step 6.1

Use the slope $\frac{1}{3}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 6.2

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

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

Step 7

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

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Step 7.1

Solve for $y$ .

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Step 7.1.1

Add $y$ and $0$ .

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

Step 7.1.2

Simplify $\frac{1}{3} \cdot (x + 0)$ .

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Step 7.1.2.1

Add $x$ and $0$ .

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

Step 7.1.2.2

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

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

Step 7.2

Reorder terms.

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

Step 8