Find the Perpendicular Line y=3x (1,3)

$y = 3 x$ $(1, 3)$

Step 1

Use the slope-intercept form to find the slope.

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Using the slope-intercept form, the slope is $3$ .

$m = 3$

Step 2

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

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

Step 3

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

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

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

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

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

Step 4

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

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x - 1)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{1}{3}$ .

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

Multiply $- \frac{1}{3} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

$y = - \frac{x}{3} + \frac{1 + 9}{3}$

Add $1$ and $9$ .

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

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{10}{3}$

Remove parentheses.

$y = - \frac{1}{3} x + \frac{10}{3}$

Step 5

Find the Perpendicular Line y=3x (1,3)

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