Solving Linear Equations

easy way to do this because i love proportions like none other. I'm not going to bother to make 100% sure this is right because I'm hard pressed for time now.

6x+5y=10
ax+2y=b
5/2=6/a=10/b (they all will follow this proporion now)
a=12/5 (if you need an explanation for this I'll give it but it's simple algebra)
b=4

soooo....
(12/5)x+2y=4 equals 6x+5y=10

Remember proportions ar you're friend. If you manipulate them right they can actually solve almost any algebra and geometry problem.

Before applying El-Diablo's proportion technique, you need to realize a few things:

1) Both equations represent lines in a plane
2) Two lines in a plane
a) are parallel (0 solutions),
b) cross or intersect at one point (1 solution), or
c) are coincident (an infinite number of solutions)

Since you know that this is case (c), you can apply El-Diablo's proportion technique.

In a coordinate plane (Cartesian coordinates) these equations represent lines and since the lines are coincident (the same as) they have an infinite number of solution for x and y. So the trick is to determine what a and b makes ax+2y=b coincident to 6x+5y=10.

If I were to do this I would multiply 6x+5y=10 through by 2/5, so this equation becomes (6*2/5)x+(5*2/5)=(10*2/5) which after reduction becomes 2.4x+2y=4. Now if this equation is to be coincident (the same as) with ax+2y=b then a=2.4 and b=4.

Rap