6/(2x - y) + 7/(3y -x)= 7

Question

4/(2x - y) + 14/(3y -x) =6

Please explain step by step to solve this equation.

Answer 1

You could clear fractions, but that would produce quadratics in x and y. An easier method might be to identify new functions, u and v:

u = 1/(2x-y)
v = 1/(3y-x)

Then the two equations become

6u + 7v = 7
4u + 14v = 6

Now if you double the first equation and subtract, you wind up with

8u = 8
So, u=1
That means v=1/7

Now, back to x and y:

1/(2x-y) = 1
1/(3y-x) = 1/7

getting rid of fractions and rearranging things a bit gives

2x-y = 1
x-3y = -7
So, we see that y=2x-1, and
x-3(2x-1) = -7
x-6x+3 = -7
x = 2
So, y = 3

You will find that (2,3) satisfies the original equations.