1.3 Consistent and Inconsistent Systems of Equations
Let us consider the fruits’ system of equations we saw earlier:
\[x + y = 20\] \[x = y\]
Each equation graphically represents a line. \(x + y = 20\) can be graphed as follows:
Also, \(x = y\) can be graphed as follows:
Graphing the two lines together, we get:
Where the intersection point \((10, 10)\) for \(x=10\) and \(y=10\) represents the solution we found earlier to the system of equations.
If the intersection point is the solution to the system of equations, then we ask about the possibilities of having solutions graphically.
As we will prove shortly, a system of equations can have only one of these three possibilities:
- No solution
- One solution
- Infinite number of solutions
These are the only possibilities. So, there is no finite number of solutions more than \(1\). Let us consider two lines in a two-dimensional (2D) Cartesian plane; The discussion forward is concerned with Euclidean Geometry.
The two lines can either be parallel or not. If they are not parallel, and going in a straight line, by definition, they must meet at one and only one point. Let us explicitly go through this exercise as this is the foundation and, we need to make it solid. Consider the \(x\)-axis as one line and another line that is not parallel to it, which we will call \(B\). If we were to pick a point \(P\) on line \(B\) and start walking alongside the line, one direction would get us closer to the x-axis and, the other direction will always get us further because they are both straight lines. Going along the direction towards the \(x\)-axis, we will hit the \(x\)-axis at one point. Once we get to our solution point -the intersection point- and keep going, the only possibility is to go further from the \(x\)-axis due to linearity. Therefore, in such case where we have two non-parallel lines, there must be one and only one solution.
This can be visualized by the following graph:
If that is not the case, meaning the two lines are not non-parallel, they can only be parallel. If two lines are parallel, they can either be separate or coincide (lie on top of each other).
If they are parallel and separate, then traveling along any line in any direction does not get us any closer to the other line. In such a case where we have two parallel lines but are separate, there is no solution. This covers the second possibility of the system of equations.
This second possibility can be visualized by the following graph:
If the two lines are parallel but are not separate, it means they represent the same line. So, all points of a line meet the other line. That means we have infinite number of solutions. This covers that last possibility.
This third and last possibility can be visualized by the following graph (if you only see one line, it is because the two lines are overlapping):
By exhaustion, we have the following theorem:
A theorem like this has implications that might not be obvious immediately, so we explicitly state them as corollaries:
A system of equations cannot have a finite number of solutions other than one solution.
If a system is found to have more than one solution, it must have infinitely many solutions.
A corollary follows logically. It might not be obvious to derive but, once stated, it should be evident.
Considering Theorem 1.1 when we say that a system of equations has a solution, does that mean it has only one solution or it has at least one solution, so possibly infinite? To avoid the ambiguity between whether a system has a solution at all, and whether it has only one or not, we call a system that has a solution or infinitely many a consistent system, and we call a system that does not have any solution an inconsistent system. Then, a consistent system of equations has one solution or infinite solutions.
So, in the next sections, we look at cases where a system can be consistent and inconsistent. We will start in the next section with a special case of a system that is always consistent.