Find the smallest solution for which all four variables are positive integers

Question

5x+ y+ z=K

x+7y+ z=K

x+ y+8z=K

a. Express Equations (5) as a homogeneous linear system of three equations in four unknowns (x, y, z, and K), and show that the solution set has one arbitrary parameter.

b. Find the smallest solution for which all four variables are positive integers.

Answer 1

a. Sabtructing equations we get: 4x - 6y = 0; 6y - 7z = 0; x = 3y/2; z = 6y/7. Substituting in last equation:

3y/2 + y + 48y/7 = K; 21y + 14y + 96y= 14K; y = 14K/131; x = 21K/131; z = 12K/131.

So, we have one parametr K

b) We have minimal positive integers solutions when K = 131, then x = 21, y = 14, z = 12