Find the Perpendicular Line (3,-2) that is perpendicular to the line 3x+4y=9

$(3, - 2)$ that is perpendicular to the line $3 x + 4 y = 9$

Step 1

Solve $3 x + 4 y = 9$ .

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

$4 y = 9 - 3 x$

Divide each term in $4 y = 9 - 3 x$ by $4$ and simplify.

Divide each term in $4 y = 9 - 3 x$ by $4$ .

$\frac{4 y}{4} = \frac{9}{4} + \frac{- 3 x}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{9}{4} + \frac{- 3 x}{4}$

Divide $y$ by $1$ .

$y = \frac{9}{4} + \frac{- 3 x}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{9}{4} - \frac{3 x}{4}$

Step 2

Find the slope when $y = \frac{9}{4} - \frac{3 x}{4}$ .

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.

$y = m x + b$

Reorder $\frac{9}{4}$ and $- \frac{3 x}{4}$ .

$y = - \frac{3 x}{4} + \frac{9}{4}$

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

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{9}{4}$

Remove parentheses.

$y = - \frac{3}{4} x + \frac{9}{4}$

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m = - \frac{3}{4}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{3}{4}}$

Step 4

Simplify $- \frac{1}{- \frac{3}{4}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{3}{4}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Multiply $- - \frac{4}{3}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$m_{perpendicular} = \frac{4}{3}$

Step 5

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

Use the slope $\frac{4}{3}$ and a given point $(3, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2) = \frac{4}{3} \cdot (x - (3))$

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

$y + 2 = \frac{4}{3} \cdot (x - 3)$

Step 6

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

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x - 3)$ .

Rewrite.

$y + 2 = 0 + 0 + \frac{4}{3} \cdot (x - 3)$

Simplify by adding zeros.

$y + 2 = \frac{4}{3} \cdot (x - 3)$

Apply the distributive property.

$y + 2 = \frac{4}{3} x + \frac{4}{3} \cdot - 3$

Combine $\frac{4}{3}$ and $x$ .

$y + 2 = \frac{4 x}{3} + \frac{4}{3} \cdot - 3$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

$y + 2 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 (- 1))$

Cancel the common factor.

$y + 2 = \frac{4 x}{3} + \frac{4}{3} \cdot (3 \cdot - 1)$

Rewrite the expression.

$y + 2 = \frac{4 x}{3} + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$y + 2 = \frac{4 x}{3} - 4$

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

Subtract $2$ from both sides of the equation.

$y = \frac{4 x}{3} - 4 - 2$

Subtract $2$ from $- 4$ .

$y = \frac{4 x}{3} - 6$

Reorder terms.

$y = \frac{4}{3} x - 6$

Step 7

Find the Perpendicular Line (3,-2) that is perpendicular to the line 3x+4y=9

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