General Solutions to a Problem

• llamaprobe5
In summary, the equation ab-a-b=x can be simplified to cd=x+1, where c and d are any two numbers whose product is x+1. For values of x such as 50, there can be multiple solutions, which can be found by finding all possible pairs of numbers whose product is x+1.
llamaprobe5
I am trying to find a general rule for finding solutions to the problem ab-a-b=x. Solving for a or b, it is apparent that 2 and 2+x always work as solutions. Some values of x, however, such as as 50, have multiple solutions. Any suggestions as to how to solve for these solutions?

You can simplify the equation a bit:
ab-a-b = x
a(b-1)-b = x
substitute c = b-1
ac-c-1 = x
c(a-1) = x+1
substitute d=a-1
cd = x+1
so any two numbers whose product equals x+1 give a solution.

for x=50:
51 = 1*51 = 51*1 = 3*17 = 17*3 = (-1*)(-51) = (-51)*(-1) = (-3)*(-17) = (-17)*(-3)
so there are 8 solutions: (2, 52), (52, 2), (4, 18), (18, 4), (0,-50), (-50,0), (-2, -16), (-16, -2)

1. What is a general solution to a problem?

A general solution to a problem is a method or set of steps that can be applied to multiple instances of a problem to achieve a satisfactory outcome. It is a broad approach that can be adapted to different situations.

2. How do you identify a general solution to a problem?

To identify a general solution to a problem, you first need to understand the underlying cause of the problem. Then, you can analyze the problem and look for common patterns or factors that are present in multiple instances of the problem. The general solution will address these common factors.

3. What are the benefits of using a general solution to a problem?

Using a general solution to a problem can save time and effort, as it eliminates the need to come up with a unique solution for each instance of the problem. It also allows for consistency and reliability in problem-solving, as the same approach can be applied to different situations.

4. Can a general solution be applied to any problem?

No, a general solution may not be applicable to all problems. Some problems may require specific and unique solutions. However, a general solution can often be adapted or modified to fit different problems.

5. How can a general solution be improved upon?

A general solution can be improved upon by gathering feedback and data from its implementation in different situations. This can help identify any flaws or limitations in the solution and allow for adjustments to be made. Additionally, keeping an open mind and continuously seeking new information and ideas can lead to improvements in the general solution.

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