# In a xy plane, the equation of a line is x+3y=12. What is an equation of a line that is perpendicular to it?

Aug 29, 2016

$y = 3 x + c$ (for all $c \in \mathbb{R}$)
For example, $y = 3 x + 4$

#### Explanation:

The equation of a straight line in slope ($m$) and intercept ($c$) form is:

$y = m x + c$

In this example we are given the equation: $x + 3 y = 12$

Rearranging terms:
$3 y = - x + 12$

$y = - \frac{1}{3} x + 4$

Hence: $m = - \frac{1}{3}$ and $c = 4$

For two straight lines to be perpendicular to eachother their slopes
(${m}_{1} \mathmr{and} {m}_{2}$) must satisfy the relationship: ${m}_{1} \times {m}_{2} = - 1$

Hence the straight lines perpendicular to the line with a slope of $- \frac{1}{3}$ will have slopes of $- \frac{1}{- \frac{1}{3}} = 3$

$\therefore$ The straight lines perpendicular to the given line will have equations:

$y = 3 x + c$ (for all $c \in \mathbb{R}$)

For example, the line perpendicular to the given line with the same intercept is $y = 3 x + 4$