Find the Perpendicular Line -3x+5y=-10 ; P=(0,-1)

$- 3 x + 5 y = - 10$ ; $P = (0, - 1)$

Solve $- 3 x + 5 y = - 10$ .

Add $3 x$ to both sides of the equation.

$5 y = - 10 + 3 x$

Divide each term in $5 y = - 10 + 3 x$ by $5$ and simplify.

Divide each term in $5 y = - 10 + 3 x$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $- 10$ by $5$ .

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

Find the slope when $y = - 2 + \frac{3 x}{5}$ .

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

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

Reorder terms.

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

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

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Use the slope $- \frac{5}{3}$ and a given point $P = (0, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = - \frac{5}{3} \cdot (x - (0))$

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

$y + 1 = - \frac{5}{3} \cdot (x + 0)$

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

Solve for $y$ .

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

Add $x$ and $0$ .

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

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

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

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

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

Subtract $1$ from both sides of the equation.

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

Reorder terms.

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

Remove parentheses.

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

Find the Perpendicular Line -3x+5y=-10 ; P=(0,-1)

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