Question

How would you describe the graph of a system of equations if the solution to the system were all real numbers

Answer 1

Let's solve this problem by using System of Linear Equations in two variables. We know that a Linear Function is given by:

$f(x)=ax+b$

that is a line with slope $m=a$ and $y-intercept$ at $(0,b)$

Both the domain and the range of this function are All Real Numbers. When the number of equations. in a system of linear equations, is the same as the number of variables there is likely to be a solution. It is not guaranteed, but likely there will be a solution or solutions.

In fact, there are three possible cases:

1. No solutions (Inconsistent System).
2. One solution (Consistent and Independent System)
3. Infinitely many solutions (Consistent and Dependent System)

We need our solution to be all real numbers, therefore our system must be Consistent and Dependent, that is, a system that has Infinitely many solutions because the two equations are really the same line. For instance:

$(1) \ x+y=3 \\ (2) \ 4x+4y=12$

Those equations are Dependent, because, in fact, the are the same equation, just multiply (1) by 4 and you will get (2). Therefore, the graph of this system is shown in the figure below. You can see that the solution is All Real Numbers, because both the domain and the range are All Real Numbers.