a,b and c are positive integers such that the simultaneous equations (a−2b)x=1, (b−2c)x=1 and x+25=c have a positive solution for x. What is the minimum value of a?
To find the minimum value of a, we need to consider the given simultaneous equations and the conditions for a positive solution.
Let's analyze the given information step by step:
1. The simultaneous equations are:
- (a - 2b)x = 1 (Equation 1)
- (b - 2c)x = 1 (Equation 2)
- x + 25 = c (Equation 3)
2. We know that these equations have a positive solution for x.
3. Let's consider Equation 3: x + 25 = c
Rearranging it, we get: x = c - 25
4. Substituting Equation 3 into Equation 1, we have:
(a - 2b)(c - 25) = 1
5. Simplifying the equation, we get:
ac - 25a - 2bc + 50b = 1
6. Now, consider Equation 2: (b - 2c)x = 1
Substituting x with c - 25, we have:
(b - 2c)(c - 25) = 1
7. Expanding and simplifying Equation 2, we get:
-2c^2 + 49c - 24b = 0
8. We have two equations involving the variables a, b, c:
ac - 25a - 2bc + 50b = 1 (Equation 4)
-2c^2 + 49c - 24b = 0 (Equation 5)
9. To determine the minimum value of a, we need to consider the extreme cases where the value of a is minimized. One such case is where c is maximized.
10. Let's assume c = x (maximized value). Substituting c = x into Equation 5:
-2x^2 + 49x - 24b = 0
11. We can solve Equation 5 to find the values of x and b in terms of a.
Using the quadratic formula, we get:
x = (49 ± sqrt(49^2 - 4*(-2)*(-24b))) / (2*(-2))
Simplifying, we get:
x = (49 ± sqrt(2401 + 192b)) / (-4)
Since we need x to be positive, we consider the positive square root:
x = (49 + sqrt(2401 + 192b)) / (-4)
12. From Equation 4, we can express b in terms of a and x:
b = (ac + 50 - 25a) / (2x)
13. Substituting the value of x from Step 11 into Equation 4 and simplifying:
b = (a((49 + sqrt(2401 + 192b)) / (-4)) + 50 - 25a) / (2((49 + sqrt(2401 + 192b)) / (-4)))
14. Simplifying the equation, we get:
b = (a + 25 + (a * sqrt(2401 + 192b)) / 64)
15. Since a, b, and c are positive integers, we must find integer solutions for a, b, and c. Thus, a, b, and c should be integers that satisfy the conditions derived from Steps 12 and 14.
16. By analyzing the equation derived in Step 14, we see that a = 39, b = 191, and c = 75 is a valid set of integer solutions.
17. Therefore, the minimum value of a is 39.
To summarize, the minimum value of a that satisfies the given conditions is 39.