Question

Classify the system and determine the number of solutions) which of the following is true of the system?

{7x - y = -11
3y = 21x + 33


consistent

inconsistent

independent

no solution

dependent

infinitely many solutions

Answer 1

Hint:
1. Linear systems that consist of two non-parallel lines have exactly one solution, at the intersection of the lines.

2. Those that are parallel have two cases, if they are parallel and distinct (i.e. not overlapping), there is no solution (the lines do not intersect anywhere), because they are inconsistent.

3. Those that are parallel but overlapping, they are consistent and have infinitely many solutions (every point on one of the lines is a solution).

How to tell them apart?
1. These are systems that have coefficients that are not proportional between the equations. Mathematically, they are "linearly independent".
Example:
5x+3y=6
3x+7y=9

2 & 3. They are characterized by proportional coefficients of the variables.
Example:
5x+2y=8
10x+4y=7
Since the coefficients of 10 and 4 in the second equation can be obtained by doubling (or multiplying by any other real number) the first equation, they coefficients are proportional.
Watch out that the equations of these types may be disguised by moving terms around, such as
5x=8-2y
10+4y=7
It's your job to move them into the same pattern before comparing.

How to tell apart cases 2 and 3?
First, match the coefficients. For the above example, double the first equation to match the coefficients of the second (do not forget to double the right-hand side as well):
10x+4y=16.........(1)
10x+4y=7...........(2)
By matching the left-hand side, the right-hand side do not match. This means we have two distinct lines, or two \(different\) equations which will never intersect. So this belongs to case 2, inconsistent, and no solution.

On the other hand, if equations (1) and (2) were to come up with the right hand side equal then the equations are consistent, and the two lines overlap. The equations are then consistent and there is an infinite number solutions. These equations are mathematically called linearly dependent.

Now you have all you need to figure out the problem, and find the answers to the question.