Algebra Examples

Through $(- 3, - 9)$ , perpendicular to $- 9 x - 5 y = 72$

Step 1

Solve $- 9 x - 5 y = 72$ .

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Step 1.1

Add $9 x$ to both sides of the equation.

$- 5 y = 72 + 9 x$

Step 1.2

Divide each term in $- 5 y = 72 + 9 x$ by $- 5$ and simplify.

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Step 1.2.1

Divide each term in $- 5 y = 72 + 9 x$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{72}{- 5} + \frac{9 x}{- 5}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{72}{- 5} + \frac{9 x}{- 5}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{72}{- 5} + \frac{9 x}{- 5}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{72}{5} + \frac{9 x}{- 5}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{72}{5} - \frac{9 x}{5}$

Step 2

Find the slope when $y = - \frac{72}{5} - \frac{9 x}{5}$ .

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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 $- \frac{72}{5}$ and $- \frac{9 x}{5}$ .

$y = - \frac{9 x}{5} - \frac{72}{5}$

Step 2.1.3

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

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Step 2.1.3.1

Reorder terms.

$y = - (\frac{9}{5} x) - \frac{72}{5}$

Step 2.1.3.2

Remove parentheses.

$y = - \frac{9}{5} x - \frac{72}{5}$

Step 2.2

Using the slope-intercept form, the slope is $- \frac{9}{5}$ .

$m = - \frac{9}{5}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{9}{5}}$

Step 4

Simplify $- \frac{1}{- \frac{9}{5}}$ to find the slope of the perpendicular line.

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Step 4.1

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

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Step 4.1.1

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{9}{5}}$

Step 4.1.2

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{9}{5}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{5}{9})$

Step 4.3

Multiply $\frac{5}{9}$ by $1$ .

$m_{perpendicular} = \frac{5}{9}$

Step 4.4

Multiply $- - \frac{5}{9}$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{5}{9})$

Step 4.4.2

Multiply $\frac{5}{9}$ by $1$ .

$m_{perpendicular} = \frac{5}{9}$

Step 5

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

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Step 5.1

Use the slope $\frac{5}{9}$ and a given point $(- 3, - 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 9) = \frac{5}{9} \cdot (x - (- 3))$

Step 5.2

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

$y + 9 = \frac{5}{9} \cdot (x + 3)$

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

Simplify $\frac{5}{9} \cdot (x + 3)$ .

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Step 6.1.1.1

Rewrite.

$y + 9 = 0 + 0 + \frac{5}{9} \cdot (x + 3)$

Step 6.1.1.2

Simplify by adding zeros.

$y + 9 = \frac{5}{9} \cdot (x + 3)$

Step 6.1.1.3

Apply the distributive property.

$y + 9 = \frac{5}{9} x + \frac{5}{9} \cdot 3$

Step 6.1.1.4

Combine $\frac{5}{9}$ and $x$ .

$y + 9 = \frac{5 x}{9} + \frac{5}{9} \cdot 3$

Step 6.1.1.5

Cancel the common factor of $3$ .

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Step 6.1.1.5.1

Factor $3$ out of $9$ .

$y + 9 = \frac{5 x}{9} + \frac{5}{3 (3)} \cdot 3$

Step 6.1.1.5.2

Cancel the common factor.

$y + 9 = \frac{5 x}{9} + \frac{5}{3 \cdot 3} \cdot 3$

Step 6.1.1.5.3

Rewrite the expression.

$y + 9 = \frac{5 x}{9} + \frac{5}{3}$

Step 6.1.2

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

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Step 6.1.2.1

Subtract $9$ from both sides of the equation.

$y = \frac{5 x}{9} + \frac{5}{3} - 9$

Step 6.1.2.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{5 x}{9} + \frac{5}{3} - 9 \cdot \frac{3}{3}$

Step 6.1.2.3

Combine $- 9$ and $\frac{3}{3}$ .

$y = \frac{5 x}{9} + \frac{5}{3} + \frac{- 9 \cdot 3}{3}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{9} + \frac{5 - 9 \cdot 3}{3}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $- 9$ by $3$ .

$y = \frac{5 x}{9} + \frac{5 - 27}{3}$

Step 6.1.2.5.2

Subtract $27$ from $5$ .

$y = \frac{5 x}{9} + \frac{- 22}{3}$

Step 6.1.2.6

Move the negative in front of the fraction.

$y = \frac{5 x}{9} - \frac{22}{3}$

Step 6.2

Reorder terms.

$y = \frac{5}{9} x - \frac{22}{3}$

Step 7