Question

If you are solving a linear system by elimination and substitution and you get a true statement, how many solutions does the system have?

Answer 1

Answer: Infinitely many solutions

Explanation:

Let's say we had this system

y = x+5

x-y = -5

Let's replace y with x+5 in the second equation

x-y = -5

x-(y) = -5

x-(x+5) = -5

x-x-5 = -5

0x-5 = -5

0-5 = -5

-5 = -5

We end up with a true statement since we get the same number on both sides. This system has infinitely many solutions. The two equations produce the same straight line graph. Effectively, the two lines overlap perfectly to intersect infinitely many times.

Notice how solving x-y = -5 for y leads to y = x+5. This is more evidence that both original equations are identical (just in different forms).