Find Any Equation Perpendicular to the Line 6x-5y-15=0

$6 x - 5 y - 15 = 0$

Step 1

Choose a point that the perpendicular line will pass through.

$(0, 0)$

Step 2

Solve $6 x - 5 y - 15 = 0$ .

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

Subtract $6 x$ from both sides of the equation.

$- 5 y - 15 = - 6 x$

Add $15$ to both sides of the equation.

$- 5 y = - 6 x + 15$

Divide each term in $- 5 y = - 6 x + 15$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 6 x + 15$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 6 x}{- 5} + \frac{15}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 6 x}{- 5} + \frac{15}{- 5}$

Divide $y$ by $1$ .

$y = \frac{- 6 x}{- 5} + \frac{15}{- 5}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{(6) x}{5} + \frac{15}{- 5}$

Divide $15$ by $- 5$ .

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

Step 3

Find the slope when $y = \frac{6 x}{5} - 3$ .

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.

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

Using the slope-intercept form, the slope is $\frac{6}{5}$ .

$m = \frac{6}{5}$

Step 4

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

$m_{perpendicular} = - \frac{1}{\frac{6}{5}}$

Step 5

Simplify $- \frac{1}{\frac{6}{5}}$ to find the slope of the perpendicular line.

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{5}{6}))$

Multiply $\frac{5}{6}$ by $1$ .

$m_{perpendicular} = - \frac{5}{6}$

Step 6

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

Use the slope $- \frac{5}{6}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

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

Step 7

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

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{5}{6} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

Combine $x$ and $\frac{5}{6}$ .

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

Move $5$ to the left of $x$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 8

Find Any Equation Perpendicular to the Line 6x-5y-15=0

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