<s>[INST] Solving set of 2 equations with 3 variables I'm working through an example and my answer is not coming out right. 
Two equations are given and then the solution is shown.
Equations:
$$\begin{aligned}20q_{1}+15q_{2}+7.5q_{3}&=10\\
q_{1}+q_{2}+q_{3}&=1\end{aligned}$$
Solution Given: 
$(X, (1/3)(1-5X), (1/3)(2+2X))$ for arbitrary $X$.
I set up a matrix and solved and my solution is
$((3/2)X+1,(-5/2)X-2,X)$
What's going on here?
 [/INST] The matrix representing the system is
$$(A|B) = \left(\begin{array}{ccc|c}
4 & 3 & 1.5 & 2\\
1 & 1 & 1 & 1
\end{array}\right) \longrightarrow \left(\begin{array}{ccc|c}
4 & 3 & 1.5 & 2\\
-1.5 & -1.5 & -1.5 & -1.5
\end{array}\right) \longrightarrow \left(\begin{array}{ccc|c}
4 & 3 & 1.5 & 2\\
2.5 & 1.5 & 0 & 0.5
\end{array}\right)$$
where we see that
$$2.5q_1 + 1.5q_2 = 0.5 \implies q_2 = \frac13 - \frac53q_1.$$
Substituting into the first equation we get
$$4q_1 + 1 - 5q_1 + 1.5q_3 = 2 \implies q_3 = \frac23 + \frac23q_1.$$
Now let $q_1 = X$ and the set of solutions is
$$\left\{\left(X, \frac13 - \frac53X, \frac23 + \frac23X\right)\colon X \in \mathbb R\right\}.$$
</s>