Does a system of linear equations always have two variables?

Linear? Yes. Keep in mind a "linear equation" has to be able to be simplified to this form: 
y = mx + b

Where "m" is a constant (such as 3), and so is "b." 

In that form you can only have two variables. If you add another variable you'd have to add the Z-Axis (go from a 2-D graph to a 3-D Graph) and it wouldn't be linear anymore. 

Now there are more types of system of equations, and it isn't unheard of to have multiple variables, but remember this VERY IMPORTANT piece of information. For every variable you have you'll need one more equation. 

If you have 1 variable (say x) you only need that equation. 
Ex: x² = 2, √(x²) = √(2), x = √(2)

If you have 2 variables (say x, and y) you need two equations to solve it. 

In a similar fashion if you have 3 variables (say x, y, and z) you need three equations to solve it. 

Hope that answers your question a bit.