Get values of multiple variables from simple additions

Question

What is the process to get the values of each variable when I only have these calculations to go on?

These are the sums:

a + b = 270
c + d = 190
a + d = 230
b + c = 230
e + a = 180
e + d = 150

I'm not even sure if this is solvable, or if there are multiple results possible. Can anybody help me with this?

Answer 1

To deduce a value for just one value, we can pick the shortest path from one variable to the other; this will only require 3 of the equations. Let me explain:

If we have an equation like $e+d=150$, we can rearrange the equation to say $d=150-e$, and then claim that $e \rightarrow d$. By doing this, we find the shortest cycle using the equations you gave:

$$a \rightarrow d \rightarrow e \rightarrow a,$$

because of the following equations:

$$a+d=230,$$ $$e+d=150,$$ $$e+a=180.$$

Just using these three, we can isolate $a$:

$$a+d=230$$ $$a+(150-e)=230$$ $$a+(150-(180-a))=230$$ $$a+150-180+a=230$$ $$2a-30=230$$ $$a=130$$

From this point on, we can just plug $a=130$ into the system and find all other values.

Answer 2

Lets name the equations in the order that you wrote them so.. equation (1) becomes a + b = 270... and e + d = 150 becomes equation (6)

Now,

Subtracting (6) from (2) you get c - e = 40 i.e. c = e + 40, lets call this equation (7) Subtracting (5) from (2) you get d - e = 50 i.e. d = e + 50, lets call this equation (8)

which mean d = c + 10 put this in 2 to get c = 90 and d = 100 With c, d figured out rest is easy. so others will be a = 130 , b = 140, e = 50