Find Any Equation Perpendicular to the Line y=3x+1

$y = 3 x + 1$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

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 3

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 4

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

Use the slope $- \frac{1}{3}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 5

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{1}{3} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

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

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

Reorder terms.

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

Remove parentheses.

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

Step 6

Find Any Equation Perpendicular to the Line y=3x+1

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