Find the Perpendicular Line (2,6) and perpendicular to 3x+5y=1

$(2, 6)$ and perpendicular to $3 x + 5 y = 1$

Step 1

Solve $3 x + 5 y = 1$ .

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

$5 y = 1 - 3 x$

Divide each term in $5 y = 1 - 3 x$ by $5$ and simplify.

Divide each term in $5 y = 1 - 3 x$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 2

Find the slope when $y = \frac{1}{5} - \frac{3 x}{5}$ .

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 $\frac{1}{5}$ and $- \frac{3 x}{5}$ .

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

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

Reorder terms.

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

Remove parentheses.

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

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

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

Step 3

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

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

Step 4

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

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

Step 5

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

Use the slope $\frac{5}{3}$ and a given point $(2, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 6

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

Solve for $y$ .

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

Rewrite.

$y - 6 = 0 + 0 + \frac{5}{3} \cdot (x - 2)$

Simplify by adding zeros.

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

Apply the distributive property.

$y - 6 = \frac{5}{3} x + \frac{5}{3} \cdot - 2$

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

$y - 6 = \frac{5 x}{3} + \frac{5}{3} \cdot - 2$

Multiply $\frac{5}{3} \cdot - 2$ .

Combine $\frac{5}{3}$ and $- 2$ .

$y - 6 = \frac{5 x}{3} + \frac{5 \cdot - 2}{3}$

Multiply $5$ by $- 2$ .

$y - 6 = \frac{5 x}{3} + \frac{- 10}{3}$

Move the negative in front of the fraction.

$y - 6 = \frac{5 x}{3} - \frac{10}{3}$

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

Add $6$ to both sides of the equation.

$y = \frac{5 x}{3} - \frac{10}{3} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{10}{3} + 6 \cdot \frac{3}{3}$

Combine $6$ and $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{10}{3} + \frac{6 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{5 x}{3} + \frac{- 10 + 6 \cdot 3}{3}$

Simplify the numerator.

Multiply $6$ by $3$ .

$y = \frac{5 x}{3} + \frac{- 10 + 18}{3}$

Add $- 10$ and $18$ .

$y = \frac{5 x}{3} + \frac{8}{3}$

Reorder terms.

$y = \frac{5}{3} x + \frac{8}{3}$

Step 7

Find the Perpendicular Line (2,6) and perpendicular to 3x+5y=1

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