I am given that;
$$x_1 + x_2 + x_3 = 75$$ $$x_1 + x_2 + x_4 = 75$$ $$x_1 + x_3 + x_4 = 75$$ $$x_2 + x_3 + x_4 = 75$$
I need to find $x_1, x_2, x_3$ and $x_4$. I know that each variable equals 25. However I am not sure how to go about showing it. Could someone explain please?
On
My Old Soulution:
well you can from $x1+x2+x3=75$ and $x1+x2+x4=75$ get that $x4=x3$ (by Subtract the equations) wich mean that $75=x1+x3+x4=x1+x3+x3$ now from $x1+x2+x3=75$ and ower statment that $x1+x3+x3=75$ we can get that $x2=x3$ (by Subtract the equations) now we will use $x2=x3$ and $x4=x3$ on $x2+x3+x4=75$ to get $x3+x3+x3=75$ now again from $x1+x3+x3=75$ and $x3+x3+x3=75$ we can find out that $x3=x1$ now that all of the varivles worse to $x3$ we got(at all of the equations) that $3*x3=75$ $x3=25$ and evry varibale worse to $x3$ whic mean $x1=x2=x3=x4=25$
~~~~~~~~~~~~~~EDIT:Found an easier solution->~~~~~~~~~~~~~~
Add all the equations:
$x1+x2+x3=75$
$+$
$x1+x2+x4=75$
$+$
$x1+x3+x4=75$
$+$
$x2+x3+x4=75$
$=$
$3x1+3x2+3x3+3x4=300$
divied by 3 and get:
$x1+x2+x3+x4=100$
as you remember we know that the sum of any 3 of the "$x$" equal 75
and $100-75=25$
This is a particular kind of system. Think of $x_1,x_2,x_3,x_4$ as being seated around a round table (in this order). Then, the system says that the sum of every three consecutive numbers is equal to $75$. That means that the fourth is always equal to the sum of all minus $75$. Therefore all your numbers are equal, and they are $25$.
This works for a general system of the form: $$x_1 + x_2 + x_3 = 75$$ $$x_2 + x_3 + x_4 = 75$$ $$ ... $$ $$ x_{n-1}+x_n+x_1 = 75$$ $$ x_n+x_1+x_2 = 75$$
To see how the solution works in the general case, it is enough to note that
$x_1=x_4=...=x_{3k+1}=...$,
$x_2 = x_5 = ... = x_{3k+2} = ...$,
$x_3 =x_6 = ... = x_{3k} = ...$.
Thus, if $n$ is not a multiple of three, then all numbers are equal. If $n$ is a multiple of three, any combination of three numbers with sum $75$ repeated periodically, is a solution.