If 3x + 2y = 10 and 2xy = 7, then find the value of 3x − 2y (if 3x − 2y 0).

Question

If 3x + 2y = 10 and 2xy = 7, then find the value of 3x − 2y (if 3x − 2y > 0).

Accepted Answer

Given:  
3x + 2y = 10  ...(1)  
2xy = 7  ...(2)

We need to find the value of 3x - 2y, given it is positive.

Step 1: Let A = 3x and B = 2y.   
Then from (1), A + B = 10.  
We want to find A - B.

Step 2: Note that (A + B)^2 = A^2 + 2AB + B^2 = 10^2 = 100.

Step 3: Also, (A - B)^2 = A^2 - 2AB + B^2.

Step 4: Subtract (A - B)^2 from (A + B)^2:   
(A + B)^2 - (A - B)^2 = 4AB.

Step 5: From (2), 2xy = 7.  
We have A = 3x and B = 2y, so AB = (3x)(2y) = 6xy.   
Given 2xy = 7, so xy = 7/2 = 3.5.   
Therefore, AB = 6 * 3.5 = 21.

Step 6: Calculate (A - B)^2:   
(A - B)^2 = (A + B)^2 - 4AB = 100 - 4*21 = 100 - 84 = 16.

Step 7: Since 3x - 2y = A - B > 0, we take the positive root:   
A - B = sqrt(16) = 4.

Answer: 4.

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