SOLUTION: Use the following: -8x+20y=-40 and 24x-60y=120 Explain what happens when you try to solve this linear system using an elimination strategy. What does this tell you about the gr

-8x + 20y = -40,    (1)
24x - 60y = 120.    (2)

I will use the Elimination method.
For it, I multiply the equation (1) by 3 (both sides). Then I get the equation

-24x + 60y = -120.   (1')


Now I add equations (1') and (2). I will get

0*x + 0*y = -120 + 120,   or

0x + 0y = 0.


This equation is ALWAYS TRUE, for any values of x and y. 


It means that the system (1'),(2) makes only ONE restriction for two unknowns x and y. 

Since the system (1'),(2) is equivalent to the system (1),(2), it means that the original system (1),(2) makes only ONE restriction 
for two unknowns x and y. 


If you look into equations (1) and (2) attentively, you will see that the second equation is PROPORTIONAL to the first equations.
Both the coefficients at x and y of the first equation are proportional to the coefficients at x and y of the second equation.
Moreover, the right hand sides are proportional with THE SAME proportionality coefficient.

It exactly means what we noticed above: the original system (1),(2) makes only ONE restriction for two unknowns x and y. 

Therefore, the system (1),(2) is EQUIVALENT to only ONE equation, EITHER (1) OT (2).
Therefore, the given system has INFINITELY MANY solutions.
You can choose x by an arbitrary way and then find the corresponding value of y from the either equation.


In this situation, the equations (1) and (2) are called DEPENDENT equations, and the system is called DEPENDENT.

A depended system has INFINITELY MANY solutions.