When would you use each method?

Question

Three methods to solving Linear Systems with two Equations. They are the Graph method, the Elimination method, and the Substitution method. What makes each method better than the other methods?

Answer 1

Hideous was not really correct. In situations where the equations are not independent, this gives the true picture, which can be useful. With this in mind, elimination gives a final result where all three possibilities are easily seen:

1) a unique answer, which is displayed.

2) an infinite number of possible answers, which are displayed (parametrically), shown by line(s) 0=0.

3) no answers, shown by a line implying that 0=1.



But there are many more ways to solve these. I believe all of these may be used:

Cramer's Rule, LU Decomposition, LQ Decomposition, QR Decomposition, Cholesky Decomposition, Singular Value Decomposition (SVD).

Answer 2

Mostly dependent on how much excess you can eliminate with each method. Elimination is good if coefficients are multiples of each other, and substitution is good if a coefficient is easily reduced to 1. Graphing mostly depends on how accurate you graph on paper (unless you're using a calculator, in which case it's dependent on how fast you can mash buttons).



If memory serves me correctly, all three will give you hideous results if the two equations have exactly 0 or infinite solutions.



Personally, I use the matrix algebra method, but that's another story.

Answer 3

Graphical is not used very much in early maths, but becomes extremely useful in Calculus. But you would never draw the lines on a piece of paper and then try to find the intersection, you'd just plug it in on the calculator and look for the intersection.



Elimination is not very flexible and is the least often used. You'd use it when you see that coefficients can be cancelled out across equations. You almost never use it, but it's nice to keep in mind just in case.



Substitution is the main algebraic method. It's mathematical and precise, and you'll fall back on this most of the time.