Modulo

[XodoX]

I forgot how I can solve this... Like this example:

Solve for the integers modulo 7:

2x − 3y = 2
x + y = 4

Can somebody please explain this to me? Thanks!

[HallsofIvy]

Solve it just the way you would an ordinary system of equations- except work "modulo 7".
You want combine the equations to eliminate one unknown so you have a single equation in the other. Multiplying the second equation by 3 will give "-3y" in one equation and "+3y" in the second so adding will eliminate y.

2x- 3y= 2
3x+3y= 5 (3*4= 12= 7+ 5 so 12= 5 (mod 7))
Adding 5x= 0 (2+ 5= 7= 0 (mod 7) or you could have left the 12: 2+ 12= 14= 2(7)= 0 (mod 7). Of course, 5x= 0 gives us x= 0.

With x= 0, the two orignal equations are now -3y= 2 and y= 4. Which are compatible because -3(4)= -12= -14+ 2= 2(mod 7).

[XodoX]

Thanks. Can I not solve them separately,too?