If x/a + y/b = 3 and x/b - y/a = 9, then what is the ratio x/y expressed in terms of a and b?

Introduction / Context:
This algebra problem involves two simultaneous equations in x and y, but the coefficients contain parameters a and b. Such questions test your ability to handle symbolic manipulation and to find a ratio like x/y without necessarily solving for the individual values of x and y numerically.

Given Data / Assumptions:

• x/a + y/b = 3
• x/b - y/a = 9
• a and b are non zero real numbers
• Required expression: x/y in terms of a and b

Concept / Approach:
We treat the two equations as a system of linear equations in x and y and solve them symbolically. Once expressions for x and y are obtained, we divide x by y to get x/y. Because a and b appear in denominators, it is convenient to clear denominators early by multiplying through by ab, which turns the equations into a more standard linear form with respect to x and y.

Step-by-Step Solution:
Step 1: Start with x/a + y/b = 3 and multiply both sides by ab to get bx + ay = 3ab. Step 2: Start with x/b - y/a = 9 and multiply both sides by ab to get ax - by = 9ab. Step 3: Now the system is bx + ay = 3ab and ax - by = 9ab. Step 4: Solve this system. Multiply the first equation by a to get abx + a^2y = 3a^2b. Step 5: Multiply the second equation by b to get abx - b^2y = 9ab^2. Step 6: Subtract the second modified equation from the first: (abx + a^2y) - (abx - b^2y) = 3a^2b - 9ab^2. Step 7: This simplifies to a^2y + b^2y = 3a^2b - 9ab^2, so y(a^2 + b^2) = 3ab(a - 3b). Step 8: Hence y = 3ab(a - 3b)/(a^2 + b^2). Step 9: Substitute y into bx + ay = 3ab to find x, or use symmetry; solving gives x = ab(3a + b)/(a^2 + b^2). Step 10: Now compute x/y = [ab(3a + b)/(a^2 + b^2)] divided by [3ab(a - 3b)/(a^2 + b^2)] = (3a + b)/(a - 3b). Step 11: Therefore x/y = (b + 3a)/(a - 3b).

Verification / Alternative check:
You can verify the ratio by choosing specific numerical values for a and b, such as a = 1 and b = 2, computing x and y from the original equations, and then checking that x/y matches the formula (b + 3a)/(a - 3b). This numeric check confirms that the symbolic derivation is correct and that the ratio is consistent for valid values of a and b where the denominators are non zero.

Why Other Options Are Wrong:
(a + 3b)/(b - 3a) is a common incorrect result that comes from sign errors when subtracting equations. (1 + 3a)/(a + 3b) ignores the structure of the coefficients involving both a and b. (a + 3b^2)/(b - 3a^2) introduces powers of a and b that do not arise in the correct elimination. (a - 3b)/(b + 3a) is another incorrect inversion of the correct ratio and does not satisfy the original system when tested with sample values.

Common Pitfalls:
Sign mistakes during elimination are very common, especially when subtracting equations with parameters. Another pitfall is cancelling terms like a^2 + b^2 incorrectly or assuming they are zero, which is generally not valid. Always keep track of coefficients carefully and avoid shortcuts that may ignore the role of both parameters.

Final Answer:
The required ratio is (b + 3a)/(a - 3b).