Solutions

   What do you think of when you see, or read, the word "solution"? Yes, it may be either in chemical equations or the answer to the problem. But solutions are applied, too, to Linear Algebra. The question is, what is a solution?

   A system of linear equations is a finite set of linear equations, each with the same variables. A solution of system of linear equations is a vector that is simultaneously a solution of each equation in the system. The solution set of a system of linear equations is the set of all solutions of the system.We will refer to the process of finding the solution set of a system of linear equations as “solving the system.”
  As I have mentioned in my previous post, there are three roads that leads to an answer, and there are three types of solutions that leads to the answer.

Solve the following systems of linear equations:
(a)x - y = 1
    x + y = 3

(b)x - y = 2
    2x - 2y = 4

(c)x - y = 1
    x - y = 3

Yes? No? Okay.
Solutions:
(a) Adding the two equations together gives 2x = 4, so x = 2, from which we find that y = 1. A quick check confirms that [2, 1] is indeed a solution of both equations. That this is the only solution can be seen by observing that this solution corresponds to the point of intersection (2, 1) of the lines with equations x + y = 1 and x + y = 3, as shown in the figure:

(b) The second equation in this system is just twice the first, so the solutions are the solutions of the first equation alone—namely, the points on the line x - y = 2. These can be represented parametrically as [2  t, t].

(c) Two numbers x and y cannot simultaneously have a difference of 1 and 3.

Thus,

• (a) is a consistent solution. Consistent solutions has at least one (1) solution.
• (c) is an inconsistent solution. Inconsistent solutions has no intersections at all/ is parallel to each other, and we cannot find the value of one of the variables
• (b) is a type of consistent solution wherein you set a parameter, so it is an infinite solution. A parameter are any counting, whole, an integer, real, rational, or irrational numbers that is used to substitute a term to limit the infinite solutions