Question

How do you know if the system `3x+2y=4` and `-2x+2y=24` is consistent or inconsistent?

Answer 1

Your system is consistent.
First of all observing the coefficients of your unknowns they are not multiples. So basically you have two lines with different slopes so they must meet somewhere and this point in common (the coordinate of it) will be the solution of your system.
You can solve the system by multiplying the first equation by `-1` and adding to the second:
You should get:
`-3x-2y=-4`
`-2x+2y=+24`
------------------ = -----------
`-5x+0=+20`
`x=-4`
Substituting in one of the equations of your system you get:
`y=8`

So your solution is `x=-4 and y=8` (which are the coordinate of the common point of your two lines).
You can check by substituting into the two equations of your system and see if you get an identity.