Comparing Methods for Solving Linear Systems

Two angles are complementary when the sum of their angles is \(\begin{align*}90^\circ\end{align*}\). Angles \(\begin{align*}A\end{align*}\) and \(\begin{align*}B\end{align*}\) are complementary angles, and twice the measure of angle \(\begin{align*}A\end{align*}\) is \(\begin{align*}9^\circ\end{align*}\) more than three times the measure of angle \(\begin{align*}B\end{align*}\). Find the measure of each angle.

First we write out our 2 equations. We will use \(\begin{align*}x\end{align*}\) to be the measure of angle \(\begin{align*}A\end{align*}\) and \(\begin{align*}y\end{align*}\) to be the measure of angle \(\begin{align*}B\end{align*}\). We get the following system:

\(\begin{align*}x + y &= 90\\ 2x &= 3y + 9\end{align*}\)

First, we'll solve this system with the graphical method. For this, we need to convert the two equations to \(\begin{align*}y = mx + b\end{align*}\) form:

\(\begin{align*}& x + y = 90 \qquad \ \Rightarrow y = -x + 90\\ & 2x = 3y + 9 \qquad \Rightarrow y = \frac{2}{3}x - 3\end{align*}\)

The first line has a slope of -1 and a \(\begin{align*}y-\end{align*}\)intercept of 90, and the second line has a slope of \(\begin{align*}\frac{2}{3}\end{align*}\) and a \(\begin{align*}y-\end{align*}\)intercept of -3. The graph looks like this:

In the graph, it appears that the lines cross at around \(\begin{align*}x = 55, y =35\end{align*}\), but it is difficult to tell exactly! Graphing by hand is not the best method in this case!