The most basic form of a linear equation is ax=b where x is a variable and a and b are constants. If this equation is real-valued, then it is almost trivial to solve. Presuming a≠0, dividing both sides by a gives that x=b/a.
Example:
This single-variable form seems almost trivial, but that is only because it is quite easy to divide by 4. Later equations will have this form, but we'll see they are slightly more complicated.
A linear equation need not be restricted to a single variable. For example, the equation
Notice that solving for y does not give us a simple constant like our previous example. Instead, it gives us the line y=-(a/b)x+(c/b) which has a solution corresponding to every possible value of x.
It is important to notice that this equation is the sum of variables times constants—not one of the variables is squared, or cubed-rooted, etc. This is what makes the equation linear.
With just one 2-D linear equation, solving for one of the variables gives you an infinite number of solutions, however, if you have two equations, we can often find one unique solution. This happens where the two lines intersect (or equivalently, when the two equations have different slopes).
Example:
When we have a list of equations that are somehow related, we call it a system. Because both these equations are linear, we call the pair
