Each of x, y, z, and u is a
positive integer that satisfies this set of equations:
- x + y = 3(z+u)
- x + z = 4(y+u)
- x + u = 5(y+z)
Determine the minimum value of x+y+z+u
Choose a variable to isolate, I choose x:
3u - y + 3z = x
4u + 4y - z = x
-u + 5y + 5z = x.
Then this is a standard linear system. Solve it however you want and you get the reduced system
u = (17/83)x
y = (7/83)x
z = (13/83)x
It is now obvious that all four variables are just scalar multiples of each other. Then the smallest solution over positive integers occurs when x=83. Then u=17, y=7, and z=13
Then the minimum value of x+y+z+u over the positive integers is 83+7+13+17=120.
Analytic Solution
