Comparing Methods for Solving Linear Systems

Finally, in this example, we'll try solving by elimination (with multiplication):

Rearrange equation one to standard form:

\(\begin{align*}& x + y = 90 \qquad \Rightarrow 2x + 2y = 180\end{align*}\)

Multiply equation two by 2:

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

Subtract:

\(\begin{align*}& \quad \qquad \qquad \qquad 2x + 2y = 180\\ & \qquad \qquad \ - \ \ (2x - 3y) = -9\\ & \qquad \qquad \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ & \quad \qquad \qquad \qquad \qquad 5y = 171\\ \\ & \text{Divide by 5 to obtain} \ y = 34.2^\circ\end{align*}\)

Substitute this value into the very first equation:

\(\begin{align*}x + 34.2 &= 90 && \mathrm{subtract \ 34.2 \ from \ both \ sides}\\ x &= 55.8^\circ\end{align*}\)

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

Even though this system looked ideal for substitution, the method of multiplication worked well too. Once the equations were rearranged properly, the solution was quick to find. You'll need to decide yourself which method to use in each case you see from now on. Try to master all the techniques, and recognize which one will be most efficient for each system you are asked to solve.