Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given y = 1/3x - 2 and y = -x -6?

Answer 1

When solving this system, we see that `1/3x-2=-x-6` (Both equal to y).

Now solving for x we get :

`1/3x+x=2-6`

`therefore x=-3`

Back substituting we get `y=-3`

So in this case, there is one unique solution `(x;y)=(-3;-3)`

It may sometimes happen that when we try to solve the equation we reach a contradiction, a statement which is always false, like for example, `1=0` and this will imply that there is no solution.

Sometimes it may happen that one variable cancels out or can always be written in terms of the other one and cancels. In this case, there are infinitely many solutions.