Find the Slope of a Perpendicular Line 3x+9y=7

$3 x + 9 y = 7$

Step 1

Rewrite in slope-intercept form.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Subtract $3 x$ from both sides of the equation.

$9 y = 7 - 3 x$

Divide each term in $9 y = 7 - 3 x$ by $9$ and simplify.

Divide each term in $9 y = 7 - 3 x$ by $9$ .

$\frac{9 y}{9} = \frac{7}{9} + \frac{- 3 x}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{7}{9} + \frac{- 3 x}{9}$

Divide $y$ by $1$ .

$y = \frac{7}{9} + \frac{- 3 x}{9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3 x$ .

$y = \frac{7}{9} + \frac{3 (- x)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$y = \frac{7}{9} + \frac{3 (- x)}{3 (3)}$

Cancel the common factor.

$y = \frac{7}{9} + \frac{3 (- x)}{3 \cdot 3}$

Rewrite the expression.

$y = \frac{7}{9} + \frac{- x}{3}$

Move the negative in front of the fraction.

$y = \frac{7}{9} - \frac{x}{3}$

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

Reorder $\frac{7}{9}$ and $- \frac{x}{3}$ .

$y = - \frac{x}{3} + \frac{7}{9}$

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{7}{9}$

Remove parentheses.

$y = - \frac{1}{3} x + \frac{7}{9}$

Step 2

Using the slope-intercept form, the slope is $- \frac{1}{3}$ .

$m = - \frac{1}{3}$

Step 3

The equation of a perpendicular line to $y = - \frac{1}{3} x + \frac{7}{9}$ must have a slope that is the negative reciprocal of the original slope.

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

Step 4

Simplify the result.

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 3$

Multiply $- - (1 \cdot 3)$ .

Multiply $3$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 3)$

Multiply $- 1$ by $3$ .

$m_{perpendicular} = 3$

Multiply $- 1$ by $- 3$ .

$m_{perpendicular} = 3$

Step 5

Find the Slope of a Perpendicular Line 3x+9y=7

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