Saturday, October 23, 2010

Using Symmetric Sums

Using Symmetric Sums
Problem Level: 4?
Given the equations:

2x+y+z+t+c=15
x+2y+z+t+c=13
x+y+2z+t+c=12
x+y+z+2t+c=11
x+y+z+t+2c=10

Find x+y+z+t+c

Solution:
Now, most people would try to cancel out some variables and manipulate the equation.  You can do that, but that will take a lot of time (and is boring and would make the problem a level 0 problem). Instead, we can find a faster method.

Let S = x+y+z+t+c

Now, we can substitute it in the equations:

x+S=15
y+S=13
z+S=12
t+S=11
c+S=10

These look much nicer than the previous ones.

We can add up all the equations
x+y+z+t+c+5S=61
S = x+y+z+t+c, so 
6S=61
and S=61/6

Answer: x+y+z+t+c=61/6

As you can see this method is not only faster, with skills, you can do it mentally.


1 comment:

  1. Hello sir,
    You have solved the question step by step and have answered it well .I will suggest all of you that you can take help from Simultaneous equation solver .It will help lot to all of you .

    ReplyDelete