Solve Maximum of a+b+c+d Without Wolfram

[Numeriprimi]

Hey!
I found one difficult equation for me...
It is:
Valid for real numbers a, b, c, d:
a + b = c + d
ad = bc
ac + bd = 1
What is the maximum of a+b+c+d?

And DON'T SAY wolfram, really no wolfram... I used it and it isn't good.
So... have you got any idea?

 

[Michael Redei]

Add $a$c to both sides of the second equation, and you get
$$
ad + ac = ac + bc
\iff
a(c+d) = (a+b)c = (c+d)c.
$$
One possibility now would be $c+d=0$, but then you'd also have $a+b=0$, and so $a+b+c+d=0$. So suppose $c+d\neq0$, then $a=c$, and from the first equation you also have $b=d$.

Now you're really left with only two unknowns, which you can use in the third equation to make $a+b+c+d=2a+2b$ minimal.

 

[Numeriprimi]

Hmm... Sorry, bud I don't understand last line. Ok, a+b+c+d=2a+2b, yes, but how I find numerical value of a+b+c+d in second variant?

 

[Michael Redei]

Your third equation says $ac+bd=1$, i.e. $a^2+b^2=1$, so if you assume $b\geq0$ (since you want $2a+2b$ to be maximal), you get $b=\sqrt{1-a^2}$. This means you need to find a value $a$ for which $2a+2b = 2a+2\sqrt{1-a^2}$ is maximal.

Do you know how to find such an $a$?

 

[CellCoree]

I can provide a solution to this problem without the use of Wolfram. First, we can simplify the given equations by substituting the first equation into the other two equations. This gives us:

a + b = c + d
a + c = 1/b
b + d = 1/c

Next, we can rearrange the equations to isolate a and d:

a = c + d - b
d = 1/c - b

Substituting these values into the first equation, we get:

(c + d - b) + b = c + d
c + d = c + d

This shows that the first equation is always true, and therefore does not provide any useful information in finding the maximum value of a+b+c+d. However, we can use the second and third equations to find the maximum value of a+b+c+d.

Substituting the value of d into the second equation, we get:

a = c + 1/c - b

Now, we can use calculus to find the maximum value of this equation. Taking the derivative with respect to b and setting it equal to 0, we get:

da/db = -1 + 1/c = 0
1/c = 1

Solving for c, we get c = 1. Substituting this value into the equation for a, we get a = 1 + 1/b. We can do the same process for the third equation with respect to b, and we get b = 1. Substituting this value into the equation for a, we get a = 1 + 1/1 = 2.

Therefore, the maximum value of a+b+c+d is 2+1+1+1 = 5. This solution was found without the use of Wolfram, using mathematical principles and techniques. I hope this helps in solving your difficult equation.