Comparing Methods for Solving Linear Systems

In this example, we'll try solving by substitution. Let's look again at the system:

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

We've already seen that we can start by solving either equation for \(\begin{align*}y\end{align*}\), so let's start with the first one:

\(\begin{align*}y = 90 - x\end{align*}\)

Substitute into the second equation:

\(\begin{align*}& 2x = 3(90 - x) + 9 && \mathrm{distribute \ the \ 3}\\ & 2x = 270 - 3x + 9 && \mathrm{add \ 3x \ to \ both \ sides}\\ & 5x = 270 + 9 = 279 && \mathrm{divide \ by \ 5}\\ & x = 55.8^\circ\end{align*}\)

Substitute back into our expression for \(\begin{align*}y\end{align*}\):

\(\begin{align*}y = 90 - 55.8 = 34.2^\circ\end{align*}\)

Angle \(\begin{align*}A\end{align*}\) measures \(\begin{align*}55.8^\circ\end{align*}\); angle \(\begin{align*}B\end{align*}\) measures \(\begin{align*}34.2^\circ\end{align*}\).