Intermediate Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 34

Determine an equation for the line that is perpendicular to $2x - 5y = 7$ and goes through $(4,3)$
a.) Write the equation in slope intercept form.
b.) Write the equation in standard form


If we transform the given line into point slope form, we have

$
\begin{equation}
\begin{aligned}
2x -5y &= 7 \\
\\
-5y &= -2x + 7 \\
\\
y &= \frac{2}{5}x - \frac{7}{5}
\end{aligned}
\end{equation}
$

Now that the line is in the form $y = mx + b$, by observation the slope is $\displaystyle m = \frac{2}{5}$. Then, the slope
of the perpendicular line is $\displaystyle m_{\perp} = \frac{-5}{2}$. By using point slope form, we have

$
\begin{equation}
\begin{aligned}
y - y_1 &= m (x - x_1)\\
\\
y - 3 &= \frac{-5}{2} (x - 4)\\
\\
y - 3 &= \frac{-5}{2} x + 10
\end{aligned}
\end{equation}
$

So, the equation in slope intercept form is
$\displaystyle y = \frac{-5}{2}x + 13$

b.) To find the equation in general form $Ax + By = C$, we need to multiply each side of the equation by $2$ to have

$
\begin{equation}
\begin{aligned}
2y &= -5x + 26 \\
\\
5x +2y &= 26
&& \text{Add } 5x
\end{aligned}
\end{equation}
$