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Solution to Linear Algebra Hoffman & Kunze Chapter 1.2 Exercise 1.2.2

Solution: Yes the two systems are equivalent. We show this by writing each equation of the first system in terms of the second, and conversely.
\begin{alignat*}{2}
3x_1+x_2&=\frac13(x_1-x_2)+\frac43(2x_1+x_2)\\
x_1+x_2 &= \frac{-1}3(x_1-x_2)+\frac23(2x_1+x_2)\\
x_1-x_2 &= (3x_1+x_2)-2(x_1+x_2)\\
2x_1+x_2&=\frac12(3x_1+x_2)+\frac12(x_1+x_2)
\end{alignat*}