System of Equations with Two variables I am stuck on the following question:
Consider the system of equations
$$2x + by = 1$$
$$3x + y  = c$$
For which values of b and c does the system have a unique solution?
For which values of b and c is the system inconsistent?
The internet tells me that the determinant cannot equal $0$ but I'm not sure how to apply this with the two constants

Make the system of equation is a matrix-vector equation
$$ \begin{vmatrix} 2 & b \\ 3 & 1 \end{vmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ c \end{bmatrix} $$
Now you can see under what conditions you have two identical equations and when you have two parallel lines (each equation being a line in the xy plane).
Edit 1
If the determinant is zero $2-3b=0$, $b=\frac{2}{3}$ the two equations have equivalent left hand size
$$\left. \begin{aligned} 2 x + \frac{2}{3} y & =1 \\ 3 x + y & = c \end{aligned} \right\} \left. \begin{aligned} \frac{3}{2} \left( 2 x + \frac{2}{3} y \right) & =\frac{3}{2} (1) \\ 3 x + y & = c \end{aligned} \right\} \begin{aligned} 3 x +  y & = \frac{3}{2} \\ 3 x + y & = c \end{aligned} $$
So your two cases are $c=\frac{3}{2}$ and $c \neq \frac{3}{2}$. 
The first makes both equations the same with infinite solutions. For example $$\begin{aligned} x & = \frac{t}{3} \\ y & = \frac{1}{2}-t \end{aligned}$$
The second makes the equations inconsistent. 
When $b \neq \frac{2}{3}$ all values of $c$ produce a result.
