Question

How do you know if `x+2y=4` and `2x+4y=5` is consistent or inconsistent?

Answer 1

Your system is inconsistent.
You can see this first by looking at it....at least for the `x` and `y` coefficients the second equation is the first multiplied by `2`!
This means that the 2 lines, that represents graphically your equations, are PARALLEL!!! The independence of the free coefficients (`4` and `5`) ensures that the two lines are not coincident. Your system cannot have solutions because your two lines will never cross.

Then you can also try to substitute one equation into the other and you'll find a CONTRADICTION.
For example:
From the first: `x=4-2y` into the second:
`2(4-2y)+4y=5`
`8-4y+4y=5`
`8=5` which is not true.

You can also try to subtract one equation from the other (multiplied by a constant):
For example:

Which is again not true!