When does a system of three linear equations with three variables have a single solution? Title says it all. I need this for a C++ program. Thanks in advance !
 A: I'm assuming your equations are in $3$ variables? If so, exactly when they are linearly independent.
If your equations are:
$$a_1 x + b_1 y + c_1 z = d_1$$
$$a_2 x + b_2 y + c_2 z = d_2$$
$$a_3 x + b_3 y + c_3 z = d_3$$
consider the matrix:
$$ \begin{pmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{pmatrix} $$
If it has determinant zero, then your planes do not intersect in a point. Either do that or Gaussian elimination, I'm not sure which has better stability. Do you need to find the point as well?
A: For a linear system of three equations, corresponding to $Ax=b$ ($A$ is a $3\times 3$ matrix), the system has a unique solution if and only if 
$rank(A)=rank([A \space{} b]=3$. 
Rank is easy to check (just find the reduced/row echelon form of the matrix and count the non-zero rows-that number is the rank of the matrix).
Equivalently you can find the determinant of $A$, in this case. For a unique solution here, it must be equal to $3$ - that implies that the columns of $A$ are linearly independent, and that hence any $b\in\mathbb{R}^3$ can be expressed uniquely (hence unique solution) as a linear combination of the columns of $A$.
