SOLUTION: I Need help with solving an equation in two variables using elimination method. Here's the sum:

11x - 7y =  xy,    (1)
 9x - 4y = 6xy.    (2)


There is more elegant solution. Let me show it.

Divide both sides of (1) by xy. Divide both sides of (2) by xy. You will get

 = 1,   (1')

 = 6.    (2')

Now introduce new variables u = , v = . Then (1') and (2') take the form

11u - 7v = 1,     (1'')
 9u - 4v = 6.     (2'')

It is usual system of two linear equations in two unknowns.
You can easily solve it using the Elimination method in its standard form. It is just simple technique, let me do not continue.

When you find u and v, do not forget to return back to the original x and y.

In order to make this approach absolutely clean, you must justify dividing by xy of equations (1) and (2). 
It means that you must accurately analyse the case xy = 0. 

If xy = 0, then  x=0  or  y=0  (or both).
If x=0, then from equations (1),(2) y=0.
If y=0, then from equations (1),(2) x=0.

Thus x=0, y=0 is one solution of (1), (2).

The other solution comes from the procedure described above.

A system such as this is solved easiest by using the SUBSTITUTION method. However, let's use ELIMINATION, since that's the required method.



11x – 7y = xy ------- eq (i)

9x – 4y = 6xy ------- eq (ii)

- 44x + 28y = - 4xy ------- Multiplying eq (i) by - 4 ----- eq (iii)

  63x - 28y =  42xy ------- Multiplying eq (ii) by 7 ------ eq (iv)

19x = 38xy ------- Adding eqs (iii) & (iv)

_____



11x - 7(½) = x(½) -------- Substituting ½ for y in eq (i)



22x - 7 = x -------- Multiplying by LCD, 2

22x - x = 7

21x = 7

x = , or 



SUBSTITUTION would have resulted in a quadratic and a second solution set:  

Yes! It is that SIMPLE!