How do you write an equation of a line passing through (3, 5), perpendicular to x - 3y = 9?

Apr 12, 2017

See the entire solution process below:

Explanation:

The equation given in the problem is in standard form. The standard form of a linear equation is: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

Where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

$\textcolor{red}{1} x - \textcolor{b l u e}{3} y = \textcolor{g r e e n}{9}$

The slope of an equation in standard form is: $m = - \frac{\textcolor{red}{A}}{\textcolor{b l u e}{B}}$. Therefore, the slope of the line represented by this equation is:

$m = \frac{- \textcolor{red}{1}}{\textcolor{b l u e}{- 3}} = \frac{1}{3}$

Let's call the slope of a line perpendicular to this line ${m}_{p}$.

${m}_{p} = - \frac{1}{m}$

Then the slope perpendicular to the line from the equation is:

${m}_{p} = - \frac{3}{1} = - 3$

Now, use the point-slope formula to find an equation for the line. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the point gives:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- 3} \left(x - \textcolor{red}{3}\right)$

We can also solve for $y$ to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y - \textcolor{red}{5} = \left(\textcolor{b l u e}{- 3} \times x\right) - \left(\textcolor{b l u e}{- 3} \times \textcolor{red}{3}\right)$

$y - \textcolor{red}{5} = - 3 x - \left(- 9\right)$

$y - \textcolor{red}{5} = - 3 x + 9$

$y - \textcolor{red}{5} + 5 = - 3 x + 9 + 5$

$y - 0 = - 3 x + 14$

$y = \textcolor{red}{- 3} x + \textcolor{b l u e}{14}$