Question

Is the system consistent or inconsistent? 2x + 3y = 5 and 4x + 6y = 19

Answer 1

Inconsistent

Answer 2

Let z = 2x+3y

So 2z = 2(2x+3y) = 4x+6y


This allows us to replace the left side of the first equation with z

and also allows us to replace the left side of the second equation with 2z


So we end up with


z = 5
2z = 19


Solve the second equation for z

2z = 19

z = 19/2


Since z = 5 and z = 19/2 at the same time, this means that this is impossible (z can't be two values at the same time)


So there are no solutions. This means that the system is inconsistent.

Answer 3

because if you multiply your LHS by 2 for the one eqn, you will get 4x + 6y for both given eqns.
So if you multiply by 2 on the LHS for the one eqn, your RHS must also be multiplied by 2 hence 5 x 2 = 10.
But for your other eqn, this is not the case as RHS = 19 and 10 is not equal to 19 thus it is inconsisent