Find the Perpendicular Line (2/5,-1) , 3x-2y=6

$(\frac{2}{5}, - 1)$ , $3 x - 2 y = 6$

Solve $3 x - 2 y = 6$ .

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

$- 2 y = 6 - 3 x$

Divide each term in $- 2 y = 6 - 3 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = 6 - 3 x$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $6$ by $- 2$ .

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

Dividing two negative values results in a positive value.

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

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

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

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

Reorder terms.

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

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

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

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

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

Simplify $- \frac{1}{\frac{3}{2}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

$y - (- 1) = - \frac{2}{3} \cdot (x - (\frac{2}{5}))$

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

$y + 1 = - \frac{2}{3} \cdot (x - \frac{2}{5})$

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

Solve for $y$ .

Simplify $- \frac{2}{3} \cdot (x - \frac{2}{5})$ .

Rewrite.

$y + 1 = 0 + 0 - \frac{2}{3} \cdot (x - \frac{2}{5})$

Simplify by adding zeros.

$y + 1 = - \frac{2}{3} \cdot (x - \frac{2}{5})$

Apply the distributive property.

$y + 1 = - \frac{2}{3} x - \frac{2}{3} (- \frac{2}{5})$

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

$y + 1 = - \frac{x \cdot 2}{3} - \frac{2}{3} (- \frac{2}{5})$

Multiply $- \frac{2}{3} (- \frac{2}{5})$ .

Multiply $- 1$ by $- 1$ .

$y + 1 = - \frac{x \cdot 2}{3} + 1 (\frac{2}{3}) \frac{2}{5}$

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

$y + 1 = - \frac{x \cdot 2}{3} + \frac{2}{3} \cdot \frac{2}{5}$

Multiply $\frac{2}{3}$ by $\frac{2}{5}$ .

$y + 1 = - \frac{x \cdot 2}{3} + \frac{2 \cdot 2}{3 \cdot 5}$

Multiply $2$ by $2$ .

$y + 1 = - \frac{x \cdot 2}{3} + \frac{4}{3 \cdot 5}$

Multiply $3$ by $5$ .

$y + 1 = - \frac{x \cdot 2}{3} + \frac{4}{15}$

Move $2$ to the left of $x$ .

$y + 1 = - \frac{2 x}{3} + \frac{4}{15}$

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

Subtract $1$ from both sides of the equation.

$y = - \frac{2 x}{3} + \frac{4}{15} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$y = - \frac{2 x}{3} + \frac{4}{15} - 1 \cdot \frac{15}{15}$

Combine $- 1$ and $\frac{15}{15}$ .

$y = - \frac{2 x}{3} + \frac{4}{15} + \frac{- 1 \cdot 15}{15}$

Combine the numerators over the common denominator.

$y = - \frac{2 x}{3} + \frac{4 - 1 \cdot 15}{15}$

Simplify the numerator.

Multiply $- 1$ by $15$ .

$y = - \frac{2 x}{3} + \frac{4 - 15}{15}$

Subtract $15$ from $4$ .

$y = - \frac{2 x}{3} + \frac{- 11}{15}$

Move the negative in front of the fraction.

$y = - \frac{2 x}{3} - \frac{11}{15}$

Reorder terms.

$y = - (\frac{2}{3} x) - \frac{11}{15}$

Remove parentheses.

$y = - \frac{2}{3} x - \frac{11}{15}$

Find the Perpendicular Line (2/5,-1) , 3x-2y=6

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