Question

Using elimination method, solve $101x+99y=499,9x+101y=501$.

Answer 1

The given system of equations is
$101x+99y=499$ (1)
$9x+101y=501$ (2)
Here, of course we could multiply equations by appropriate numbers to eliminate one of the variables. However, note that the coefficient of x in one equation is equal to the coefficient of y in the other equation. In such a case, we add and subtract the two equations to get a new system of very simple equations having the same solution.
Adding (1) and (2), we get $200x+200y=1000$.
Dividing by 200 we get, x+y = 5 (3)
Subtracting (2) from (1), we get $2x-2y=-2$which is same as
$x-y=-1$ (4)
Solving (3) and (4), we get x = 2, y = 3.
Thus, the required solution is ( 2, 3 ).