Find the Perpendicular Line 5x+3y=0 , (7/8,3/4)

$5 x + 3 y = 0$ , $(\frac{7}{8}, \frac{3}{4})$

Step 1

Solve $5 x + 3 y = 0$ .

Subtract $5 x$ from both sides of the equation.

$3 y = - 5 x$

Divide each term in $3 y = - 5 x$ by $3$ and simplify.

Divide each term in $3 y = - 5 x$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 2

Find the slope when $y = - \frac{5 x}{3}$ .

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.

$y = m x + b$

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

Reorder terms.

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

Remove parentheses.

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

Using the slope-intercept form, the slope is $- \frac{5}{3}$ .

$m = - \frac{5}{3}$

Step 3

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

$m_{perpendicular} = - \frac{1}{- \frac{5}{3}}$

Step 4

Simplify $- \frac{1}{- \frac{5}{3}}$ to find the slope of the perpendicular line.

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

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

$m_{perpendicular} = - \frac{- 1 \cdot - 1}{- \frac{5}{3}}$

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{\frac{5}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 (\frac{3}{5})$

Multiply $\frac{3}{5}$ by $1$ .

$m_{perpendicular} = \frac{3}{5}$

Multiply $- - \frac{3}{5}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{3}{5})$

Multiply $\frac{3}{5}$ by $1$ .

$m_{perpendicular} = \frac{3}{5}$

Step 5

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

Use the slope $\frac{3}{5}$ and a given point $(\frac{7}{8}, \frac{3}{4})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{3}{4}) = \frac{3}{5} \cdot (x - (\frac{7}{8}))$

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

$y - \frac{3}{4} = \frac{3}{5} \cdot (x - \frac{7}{8})$

Step 6

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

Solve for $y$ .

Simplify $\frac{3}{5} \cdot (x - \frac{7}{8})$ .

Rewrite.

$y - \frac{3}{4} = 0 + 0 + \frac{3}{5} \cdot (x - \frac{7}{8})$

Simplify by adding zeros.

$y - \frac{3}{4} = \frac{3}{5} \cdot (x - \frac{7}{8})$

Apply the distributive property.

$y - \frac{3}{4} = \frac{3}{5} x + \frac{3}{5} (- \frac{7}{8})$

Combine $\frac{3}{5}$ and $x$ .

$y - \frac{3}{4} = \frac{3 x}{5} + \frac{3}{5} (- \frac{7}{8})$

Multiply $\frac{3}{5} (- \frac{7}{8})$ .

Multiply $\frac{3}{5}$ by $\frac{7}{8}$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{3 \cdot 7}{5 \cdot 8}$

Multiply $3$ by $7$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{21}{5 \cdot 8}$

Multiply $5$ by $8$ .

$y - \frac{3}{4} = \frac{3 x}{5} - \frac{21}{40}$

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

Add $\frac{3}{4}$ to both sides of the equation.

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3}{4}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3}{4} \cdot \frac{10}{10}$

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{4}$ by $\frac{10}{10}$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3 \cdot 10}{4 \cdot 10}$

Multiply $4$ by $10$ .

$y = \frac{3 x}{5} - \frac{21}{40} + \frac{3 \cdot 10}{40}$

Combine the numerators over the common denominator.

$y = \frac{3 x}{5} + \frac{- 21 + 3 \cdot 10}{40}$

Simplify the numerator.

Multiply $3$ by $10$ .

$y = \frac{3 x}{5} + \frac{- 21 + 30}{40}$

Add $- 21$ and $30$ .

$y = \frac{3 x}{5} + \frac{9}{40}$

Reorder terms.

$y = \frac{3}{5} x + \frac{9}{40}$

Step 7

Find the Perpendicular Line 5x+3y=0 , (7/8,3/4)

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