Question

Find the greatest possible value for a+b+c+d if b is a positive integer and a,b,c,d satisfy the system of equations

a+b=c

b+c=d

c+d=a

Answer 1

answer is a+b+c+d= -5.b

Answer 2

`a+b+c+d = -5\cdot b`

Explanation:

The given system of equations is now reduced:

1)
`a+b = c`,
`b+c = d`,
`c+d=a`,
`b > 0` Given

2)
`a = c+d` By 1)

3)
`(c+d)+b=c` 2) in 1)

4)
`d = b + c` By 1)

5)
`[c+(b+c)]+b = c` 4) in 3)

6)
`2\cdot b + 2\cdot c = c` Algebra

7)
`c = -2\cdot b` Algebra

8)
`d = -b` 7) in 4)/Algebra

9)
`a = -3\cdot b` 7) and 8) in 2)/Algebra

10)
`a+b+c+d = -5\cdot b` 9), 1), 7) and 8)/Algebra/Result