without graphing or using any algebraic methods determine whether or not there is a solution to the following
3x+y=2
6x-2y=3
If you have asystem of linear equations like:
$\displaystyle \left|\begin{array}{l} a_1 x+b_1 y = c_1 \\a_2 x + b_2 y = c_2\end{array}\right.$
then there exist a unique solution if the determinant
$\displaystyle D=\left|\begin{array}{cc}a_1 & b_1 \\a_2 & b_2 \end{array} \right| \neq 0$
With your example $\displaystyle D = -12 \neq 0$ and therefore there must be an unique solution.