Find the Perpendicular Line through (5,-2) ; perpendicular to x-3y=3

through $(5, - 2)$ ; perpendicular to $x - 3 y = 3$

Step 1

Solve $x - 3 y = 3$ .

Subtract $x$ from both sides of the equation.

$- 3 y = 3 - x$

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $3$ by $- 3$ .

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

Dividing two negative values results in a positive value.

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

Step 2

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

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

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

Reorder terms.

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

Using the slope-intercept form, the slope is $\frac{1}{3}$ .

$m = \frac{1}{3}$

Step 3

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

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

Step 4

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $- (1 \cdot 3)$ .

Multiply $3$ by $1$ .

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

Multiply $- 1$ by $3$ .

$m_{perpendicular} = - 3$

Step 5

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

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

$y - (- 2) = - 3 \cdot (x - (5))$

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

$y + 2 = - 3 \cdot (x - 5)$

Step 6

Solve for $y$ .

Simplify $- 3 \cdot (x - 5)$ .

Rewrite.

$y + 2 = 0 + 0 - 3 \cdot (x - 5)$

Simplify by adding zeros.

$y + 2 = - 3 \cdot (x - 5)$

Apply the distributive property.

$y + 2 = - 3 x - 3 \cdot - 5$

Multiply $- 3$ by $- 5$ .

$y + 2 = - 3 x + 15$

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

Subtract $2$ from both sides of the equation.

$y = - 3 x + 15 - 2$

Subtract $2$ from $15$ .

$y = - 3 x + 13$

Step 7

Find the Perpendicular Line through (5,-2) ; perpendicular to x-3y=3

Related Questions

Support LCMGCF.com by disabling your adblocker.