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2x+y =10 (*)

3x+4y=25 (**)

By peforming 4(*) - (**), we have 5x =15 i.e. x=3. Now my question is this:

Why does the same y value (y=4) satisfy both original equations? This always happens. We never find that, having found x, the equations give 2 different values of y and we conclude there is no solution. In other words, if the equations are inconsistent, then the contradiction is derived immediately.