Solve the simultaneous equations..... 3x - y + 3 = 11 & 2x^2 + y^2 + 3 = 102 where X and Y are both positive integers.

Here we have two equations with two unknowns, the method we use to solve this is substitution. First, find one of the unknowns in terms of the other by rearranging the first equation to arrive at y = 3x - 8, then, substitute this into the second equation. Then expand the brackets and collate like terms and you arrive at 11x^2 - 48x + 67 = 102. By equating to zero you get.... 11x^2 - 48x - 35 = 0. This should now be recognised as a quadratic equation you can solve by factorising.... (11x + 7 ) (x - 5) = 0. Therefore x is either 5 or -7/11.However we must remember the question states X and Y are both positive thus we reject X = - 7/11. Finally take X = 5 and substitute it back into either of the original equations and solve for Y. You'll find that Y = 7.