Finite Math Examples

$\frac{1}{2} x + \frac{1}{3} y = - \frac{13}{6}$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $\frac{1}{2} x + \frac{1}{3} y = - \frac{13}{6}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{x}{2} + \frac{1}{3} y = - \frac{13}{6}$

Step 2.1.2

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

$\frac{x}{2} + \frac{y}{3} = - \frac{13}{6}$

Step 2.2

Subtract $\frac{x}{2}$ from both sides of the equation.

$\frac{y}{3} = - \frac{13}{6} - \frac{x}{2}$

Step 2.3

Multiply both sides of the equation by $3$ .

$3 \frac{y}{3} = 3 (- \frac{13}{6} - \frac{x}{2})$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.1

Cancel the common factor.

$3 \frac{y}{3} = 3 (- \frac{13}{6} - \frac{x}{2})$

Step 2.4.1.1.2

Rewrite the expression.

$y = 3 (- \frac{13}{6} - \frac{x}{2})$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $3 (- \frac{13}{6} - \frac{x}{2})$ .

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Step 2.4.2.1.1

Apply the distributive property.

$y = 3 (- \frac{13}{6}) + 3 (- \frac{x}{2})$

Step 2.4.2.1.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.2.1

Move the leading negative in $- \frac{13}{6}$ into the numerator.

$y = 3 (\frac{- 13}{6}) + 3 (- \frac{x}{2})$

Step 2.4.2.1.2.2

Factor $3$ out of $6$ .

$y = 3 \frac{- 13}{3 (2)} + 3 (- \frac{x}{2})$

Step 2.4.2.1.2.3

Cancel the common factor.

$y = 3 \frac{- 13}{3 \cdot 2} + 3 (- \frac{x}{2})$

Step 2.4.2.1.2.4

Rewrite the expression.

$y = \frac{- 13}{2} + 3 (- \frac{x}{2})$

Step 2.4.2.1.3

Multiply $3 (- \frac{x}{2})$ .

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Step 2.4.2.1.3.1

Multiply $- 1$ by $3$ .

$y = \frac{- 13}{2} - 3 \frac{x}{2}$

Step 2.4.2.1.3.2

Combine $- 3$ and $\frac{x}{2}$ .

$y = \frac{- 13}{2} + \frac{- 3 x}{2}$

Step 2.4.2.1.4

Simplify each term.

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Step 2.4.2.1.4.1

Move the negative in front of the fraction.

$y = - \frac{13}{2} + \frac{- 3 x}{2}$

Step 2.4.2.1.4.2

Move the negative in front of the fraction.

$y = - \frac{13}{2} - \frac{3 x}{2}$

Step 2.5

Reorder $- \frac{13}{2}$ and $- \frac{3 x}{2}$ .

$y = - \frac{3 x}{2} - \frac{13}{2}$

Step 3

Find the slope when $y = - \frac{3 x}{2} - \frac{13}{2}$ .

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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.

$y = - (\frac{3}{2} x) - \frac{13}{2}$

Step 3.1.2.2

Remove parentheses.

$y = - \frac{3}{2} x - \frac{13}{2}$

Step 3.2

Using the slope-intercept form, the slope is $- \frac{3}{2}$ .

$m = - \frac{3}{2}$

Step 4

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{2}}$

Step 5

Simplify $- \frac{1}{- \frac{3}{2}}$ 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 $- 1 (- 1)$ .

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{2}}$

Step 5.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{2}}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{2}{3})$

Step 5.3

Multiply $\frac{2}{3}$ by $1$ .

$m_{perpendicular} = \frac{2}{3}$

Step 5.4

Multiply $- - \frac{2}{3}$ .

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Step 5.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{2}{3})$

Step 5.4.2

Multiply $\frac{2}{3}$ by $1$ .

$m_{perpendicular} = \frac{2}{3}$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

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 $0$ .

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

Step 7.1.2

Simplify $\frac{2}{3} \cdot (x + 0)$ .

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Step 7.1.2.1

Add $x$ and $0$ .

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

Step 7.1.2.2

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

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

Step 7.2

Reorder terms.

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

Step 8