A math textbook gives the following problem: What is the solution of 3(2x-1) – 2(3x+4) = 11x. The solution is an integer.
Using Excel to choose some numbers at random and compute the remaining numbers, I want to create problems like that one, with coefficients and solutions that are integers. Rightly or wrongly, I’ve conceptualized my task like this:
- a(bx + c) + d(ex + f) = gx
- abx + ac + dex + df = gx
- ac + df = - abx - dex + gx
- ac + df = (-ab - de + g)x
- (ac + df) / (-ab - de + g) = x
So if we arbitrarily choose values of a, b, d, e, g and x, we can solve for the numerator. For example, if a=2, b=3, d=5, e=7, and g=11, then the denominator is -30. If we choose x=3, then the numerator is 3*(-30) = -90.
- (ac + df) = -90
- 2c + 5f = -90
Somehow I’m stuck on that last step. How do I pick values of c and f so that this works? Of course I don’t want just an example of values of c and f that solve here, but a general approach that will give values that are integers.
Or can you suggest a better approach to the original problem?