Explanation:
We have the system of equations:
1/x + 2/y = 1 -----(1)
-1/x + 8/y = -6 -----(2)
We can use the substitutions u = 1/x and v = 1/y to rewrite the equations in terms of u and v. Substituting, we get:
u + 2v = 1 -----(3)
-u + 8v = -6 -----(4)
Now we can solve the system in terms of u and v. Adding equations (3) and (4) gives:
2v = -5
Dividing both sides by 2, we get:
v = -5/2
Substituting this value of v into equation (3) gives:
u + 2(-5/2) = 1
Simplifying:
u - 5 = 1
u = 6
Therefore, we have u = 6 and v = -5/2.
To find the solution set in terms of x and y, we substitute back:
u = 1/x, so 6 = 1/x, and x = 1/6
v = 1/y, so -5/2 = 1/y, and y = -2/5
Therefore, the solution set to the original system is x = 1/6 and y = -2/5.