Find Any Equation Perpendicular to the Line 3y=5x-1

$3 y = 5 x - 1$

Choose a point that the perpendicular line will pass through.

$(0, 0)$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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 terms.

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

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

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Use the slope $- \frac{3}{5}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{3}{5} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

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

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

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

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

Reorder terms.

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

Remove parentheses.

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

Find Any Equation Perpendicular to the Line 3y=5x-1

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