Question

Which is the point of intersection when the system of equations below is graphed on the coordinate plane?

Answer 1

Since, you have not included the system, I will help you with an explanation that will teach you how to work this kind of problems, for a given system of linear equations.

For a system of two linear equations there are three possibilities:

i) No solution
ii) Infinite solutions
iii) One solution.

1) No solution:

This happens when the two linear equations represent parallel lines.

For example: 
x + y = 12 and x + y = 14.

If you subtract one equation from the other you get:

x + y - x - y = 14 - 12

0 = 2, which is absurd, meaning there is not solution. This is true whenever the two equations represent parallel lines.

That is, there is not intersection point in the graph.


2) Infinite solutions:

This situation arises when the two equations result being the same.

For example:

3x + 8y = 24 and - 16y - 6x = - 48

To solve that system, you can multiply the first equation by 2:

6x + 16y = 48

Now you can add this new equation with the second original equation:

6x - 16y = 48
-6x + 16y = - 48----------------------
0 + 0 = 0, which is alwasy true, meaning that there are infinite solutions.

The two lines overlap entirely because they are the same line.


3) One solution.


The third case, is the general case when two functions can intersect and then the intersection is the solution of the system.

For example: x + y = 12  and 2x + y = 14

The solution is:

i) multiply the first equation by 2:

2x + 2y = 24

ii) Subtract the second equation:
2x + 2y = 242x + y  = 14------------------
        y = 10

⇒ 2x = 14 - y = 14 - 10 = 4

⇒ x = 4/ 2 = 2

Then, the solution of this system is (2, 10), and that is the intersection point in the graph.