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let z = 5x+2y

double both sides
2z = 2(5x+2y)
2z = 10x+4y

so if
5x + 2y = 7
10x + 4y = 14
then we can also say
z = 7
2z = 14
where z = 5x+2y

but....what happens when we solve 2z = 14 for z?

2z = 14 ---> z = 7
but z = 7 in the first equation

so no matter what happens, we're going to get z = 7

If you got a nonsense statement such as 0 = 3, that would indicate no solution.

that makes more sense, I feel so silly now for confusing that.