Matrix Methods or Elimination

[zeamju]

My textbook is confusing me more than it's helping :-(

Here is what is described:

-0.35S[sub:5o4bc6ne]1[/sub:5o4bc6ne] +0.20S[sub:5o4bc6ne]2[/sub:5o4bc6ne] + 0.15S[sub:5o4bc6ne]3[/sub:5o4bc6ne] = 0
0.10S[sub:5o4bc6ne]1[/sub:5o4bc6ne]-0.45S[sub:5o4bc6ne]2[/sub:5o4bc6ne] + 0.25S[sub:5o4bc6ne]3[/sub:5o4bc6ne] = 0
0.25S[sub:5o4bc6ne]1[/sub:5o4bc6ne] + 0.25S[sub:5o4bc6ne]2[/sub:5o4bc6ne] - 0.40S[sub:5o4bc6ne]3[/sub:5o4bc6ne] = 0
S[sub:5o4bc6ne]1[/sub:5o4bc6ne] + S[sub:5o4bc6ne]2[/sub:5o4bc6ne] + S[sub:5o4bc6ne]3[/sub:5o4bc6ne] = 1

I already know the answers, not looking for anyone to do my homework for me.

S[sub:5o4bc6ne]1[/sub:5o4bc6ne] = 0.329
S[sub:5o4bc6ne]2[/sub:5o4bc6ne] = 0.287
S[sub:5o4bc6ne]3[/sub:5o4bc6ne] = 0.385

I don't understand where these numbers are coming from. I'd really appreciate it if someone could walk me through what is going on and how the solutions were determined. I have a test coming up this week and I really need to understand this. Thanks in advance!

 

[Denis]

-0.35S[sub:2z9t4ni8]1[/sub:2z9t4ni8] +0.20S[sub:2z9t4ni8]2[/sub:2z9t4ni8] + 0.15S[sub:2z9t4ni8]3[/sub:2z9t4ni8] = 0
0.10S[sub:2z9t4ni8]1[/sub:2z9t4ni8]-0.45S[sub:2z9t4ni8]2[/sub:2z9t4ni8] + 0.25S[sub:2z9t4ni8]3[/sub:2z9t4ni8] = 0
S[sub:2z9t4ni8]1[/sub:2z9t4ni8] + S[sub:2z9t4ni8]2[/sub:2z9t4ni8] + S[sub:2z9t4ni8]3[/sub:2z9t4ni8] = 1

Well, one of 1st 3 equations not required; so we'll go with the above;
I'm changing your 3 variables to a,b,c : why suffer with subscripts!

Multiply 1st 2 equations by 100 to get rid of them annoying decimals:
-35a + 20b + 15c = 0 [1]
10a -45b + 25c = 0 [2]
And we have:
a + b + c = 1 : so a = 1 - b - c [3]

Substitute [3] in [1]:
-35(1 - b - c) + 20b + 15c = 0 ; simplify to get:
11b = -10c + 7 [4]

Substitute [3] in [2]:
10(1 - b - c) - 45b + 25c = 0 ; simplify to get:
11b = 3c + 2 [5]

From [4] and [5]: -10c + 7 = 3c + 2 ; simplify to get:
c = 5/13 or .3846...

Using [5]: 11b = 3(5/13) + 2 ; simplify to get:
b = 41/143 or .2867...

Using [3]: a = 1 - 41/143 - 5/13 : I'll let you put on the finishing touches :wink:

 

[zeamju]

I appreciate your response! You lost me here...

-35a + 20b + 15c = 0 [1]
10a -45b + 25c = 0 [2]

Where are the [1] and the [2] coming from? Also when discarding one of the equations, how do you determine which one to get rid of? Does it even matter/