Find the Perpendicular Line Through (3,5) ; perpendicular to 2x-3y=6

Through $(3, 5)$ ; perpendicular to $2 x - 3 y = 6$

Step 1

Solve $2 x - 3 y = 6$ .

Subtract $2 x$ from both sides of the equation.

$- 3 y = 6 - 2 x$

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $6$ by $- 3$ .

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

Dividing two negative values results in a positive value.

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

Step 2

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

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.

$y = m x + b$

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

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

Reorder terms.

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

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

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

Step 3

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

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

Step 4

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Step 5

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

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

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

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

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

Step 6

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

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

Multiply $- \frac{3}{2} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

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

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

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

Multiply $3$ by $3$ .

$y - 5 = - \frac{x \cdot 3}{2} + \frac{9}{2}$

Move $3$ to the left of $x$ .

$y - 5 = - \frac{3 x}{2} + \frac{9}{2}$

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

Add $5$ to both sides of the equation.

$y = - \frac{3 x}{2} + \frac{9}{2} + 5$

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{9}{2} + 5 \cdot \frac{2}{2}$

Combine $5$ and $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{9}{2} + \frac{5 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{9 + 5 \cdot 2}{2}$

Simplify the numerator.

Multiply $5$ by $2$ .

$y = - \frac{3 x}{2} + \frac{9 + 10}{2}$

Add $9$ and $10$ .

$y = - \frac{3 x}{2} + \frac{19}{2}$

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{19}{2}$

Remove parentheses.

$y = - \frac{3}{2} x + \frac{19}{2}$

Step 7

Find the Perpendicular Line Through (3,5) ; perpendicular to 2x-3y=6

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