How do I classify systems as consistent, inconsistent and dependant or independant?

Question

I have to review for an assessment test for college and I have absolutely no rememberance of what these mean. Does consistent mean they have the same solution times infinity?

Some questions I'm working on:

Question 1:
x + 2y = 9
2x - 3y = 4

Question 2:
x - 9 = y
y+ 2 = x

Question 3:
x + 3y = 20
x + 6y = 40






Just curious as to how I solve and determine what they are. Do I use substitution or elimination? Any help would be greatly appreciated. Thanks.

Answer 1

Most systems of linear equations in two variables are of the form:

ax + by + c = 0
px + qy + r = 0

If you say a system is consistent, it means that it can be solved.
If you say a system is inconsistent, it means that the system has no solution.
If a system is dependent, it has infinite solutions and if it is independent, it has a unique solution. Both dependent and independent systems are consistent.

Now, let's see the conditions for each of these:


If:

1) a/p = b/q = c/r
then the equations are consistent and dependent. Such systems have an infinite number of solutions.

2) a/p ≠ b/q
then the system is consistent and independent

3) a/p = b/q ≠ c/r
then the system is inconsistent


The best solving method is still under dispute, as there are examples of problems which bring out the disadvantages of each method. But generally, cross-multiplication and elimination methods are preferred.

Feel free to drop me a note if you have any doubts.

Answer 2

Try to solve for x and y by the usual methods. If you end up eliminating both variables and have a true statement, like 0 = 0, then there are infinitely many solutions. The system is dependent. If you end up with a false statement like 0 = 1, then there are no solutions. The system is inconsistent. Otherwise you have independent equations and there is one solution.