Question

Use the substitution u = 1/x and v = 1/y to rewrite the equations in the system in terms of the variables u and v. Solve the system in terms of u and v. Then substitute to determine the solution set to the original system in terms of x and y.

1/x + 2/y = 1
-1/x + 8/y = -6

Answer 1

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.