First, we accept the agreement that we consider only positive values of the square roots.
After that, we solve the problem in several steps.
Step 1. We find 6x + 5y from the first equation.
= 72.
For it, we introduce new variable u = .
Then the equation takes the form
u^2 + u = 72
u^2 + u - 72 = 0
(u+9)*(u-8) = 0.
The roots are -9 and 8, but we accept only positive root u= 8.
Then we have THIS equation
6x + 5y = , or
6x + 5y = 64. (1)
Step 1 is complete.
Step 2. We find 3x - 4y from the second equation.
= 30.
For it, we introduce new variable v = .
Then the equation takes the form
v^2 + v = 30
v^2 + v - 30 = 0
(v+6)*(v-5) = 0.
The roots are -6 and 5, but we accept only positive root v= 5.
Then we have THIS equation
3x - 5y = , or
3x - 5y = 25. (2)
Step 2 is complete.
Step 3. Solving the system (1), (2) to find x and y.
The system is
6x + 5y = 64. (1)
3x - 5y = 25. (2)
Solve it by ANY WAY you know / (you like) / (you prefer).
The solution is x = ; y = .
Step 4. Calculate y - x.
It is y - x = - = = -8.
The solution is completed.
The ANSWER is y - x = -8.
