Expert: Michael Tadesse - 9/23/2009

Question
Hi Michael,

I have progressed from triple to quintuple and would appreciate the logic of working out to enable me to progress and do the rest myself. Here goes;

a+b+c+d+e=1
2a-2b+2c-2d+e=1
a+b+2c+2d+2e=1
4a-2b+4c+4d-3e=-6
a+3b-3c-d+6e=10

Many thanks in advance.

Ria

Answer
Hello Ria

I’m very astonished you are working hard and progressing. I want you to exude your confidence and strive to achieve your goals and I am there to help.

I reckon this is going to be long but grasp the concept not the calculation and you will be able to do it by yourself.


Here is what you do. You have 5 equations and 5 variables, the problem is therefore, determinate, we can find the values of all the required variables.

To proceed you have to reduce the number of variables as much as you can.

You can see that the first two equations are equal because their results are 1


A + B + C +D = 2A – 2B + 2C – 2D

E is canceled out. Check it.
Collect like terms

3B + 3D = A + C  

A = 3B + 3D – C ------------------------- The first important reduced equation

Also solve for E in term of the other 4 Variables.

E = 1 – (A + B + C + D) ------------------------ #2
Do you understand my steps Ria?

Now, insert the expressed value of A and E in the third equation.

Operate carefully and collect like terms;

C + 1 = 4B + 3D ---------------------------- #3


Similarly, insert the expressed value of A and E in the fourth equation and operate carefully and finally you’ll get;

22B + 28D = 9 ---------------------------- #4

Do the same in the fifth equation

-4 = 18B + 22D + 4C ----------------------- #5

Have you observed what I have just done? I have just reduce the number of variables from 5 to 3 (equations 3, 4 and 5) which is way easier than the quintuple.

Now, Ria, arrange equations 3, 4 and 5

C + 1 = 4B + 3D  
22B + 28D = 9
-4 = 18B + 22D + 4C

As you know, you have to write one of the three variables in terms of the other two and substitute.


B = (9 – 28D)/22
You can substitute this equation in the third equation and you will be left with only D and C, so you can solve this equation with the first …. Blah, blah …… You know it.


Finally, these are the luckiest numbers

A = 2473/616
B = -135/44
C = -3127/616
D = 153/616
E = 1477/616

Check them if you like.
Ria, I encourage you to go further, to read more and ask more. And believe in yourself that you can succeed. I will be there to assist you whenever I am free

Michael Tadesse