$8$ linear equations and $9$ unknowns can we solve?

Question

Here are the equations: \begin{align} & A+B+C=15 \tag{1} \\ & P+Q+R=15 \tag{2} \\ & X+Y+Z=15 \tag{3} \\ & A+Q+Z=15 \tag{4} \\ & C+Q+X=15 \tag{5} \\ & A+P+X=15 \tag{6} \\ & B+Q+Y=15 \tag{7} \\ & C+R+Z=15 \tag{8} \end{align} Also, provide the method in detail to solve these equations

Thanks!

Edit 1: These equations are $3 \times 3$ square, where each row/column/diagonal sum to $15$.

Edit 2: All unknowns are unique from 1 to 9.

Answer 1

By looking at the symmetry of the problem (each equation is the sum of three variables equals $15$), one solution is $$A=B=C=P=Q=R=X=Y=Z=5$$

There are infinitely many solutions, however, as the number of variables exceeds the number of equations.

---

Update, due to edits. Subtracting equation (5) from (4) yields $A-C=0$, or $A=C$. Hence it is impossible for the nine variables to each stand for distinct values from $1$ to $9$.