# Exploring Methods of Solving a Linear System

Solving a linear system is to find a point where two linear question intercept, there are two methods; one by substitution and one by elimination, so which method proves to be the more efficient and more easily done in a short amount of time?
Substitution requires the variable y to be isolated from the equation which is usually in y=mx+b form where in some cases it isn’t always in that order and requires one to perform the procedure to isolate before substituting the equation. An extra step that many students find harder to do when there is a time limit in addition to having to substitute the equations twice to get values of x and y in order to find the point of intersection or known as the solution in some schools.
The method of elimination proves to have it’s strong points and is the most preferred by most people where it not only eliminate one of the variables but also solves for the remaining variable left in the equation all in one step. So it really only requires one substitution in the entire procedure, however it has it’s limits and makes substitution more reasonable way to solving the linear system as they need the variable that is to eliminated to have the same co-efficient one positive and another which is negative(oppostite co-efficient)in order to eliminate the entire variable. In many cases the two linear equations to not have variables with the same co-efficients and require to multiply the entire equation in order for that variable same as the other one that of the other linear equation. This can be very time consuming, sometimes both linear equations need to multiply in order for the co-efficients to be the same value and to eliminate each other. While for substitution methods, there is none of that required.
For some students, they prefer either way and do not mind the extra work perhaps because they are more strong in multiplying or subtracting then they will work on the method of elimination just as fast as one who works with the method of substitution. It really depends on which one works better with one’s brain and though both result to the same answer, it is much more efficient then trying to draw a graph as the linear equations may be so apart from each other that graphs can be extended so far that a simple piece of paper cannot point out the intersection point. In addition, some points of intersection may be in fraction form and a simple graph will not be able to provide that exact answer unless the point of intersection can be in decimal form. So, the methods of substitution and elimination are ways to find the answer quickly and efficiently.
Furthermore, though through many calculations and substitutions/eliminations that it is easy to mix up and blow on some of the procedures and end up with the incorrect answer. It can be fixed with lots of practice and in the end of each procedure, students are required to check their answers with the famous left side and right side chart so it guarantees that one will be able to find the right answer and if not one can go back and probably find out what happened and fix it quickly without always restarting the entire procedure. Therefore the methods of elimination and substitution are the most effective not unless you have a graphing calculator but in the reality when solving problems requires pencil and paper the two methods are more easy and efficient to solving linear equations!