SOLUTION: Determine the augmented matrix:p+s+c=600; 12p+10s+8c=5900; 0-s+c=175 Solve for the matrix and please show all work.

Algebra ->  Algebra  -> Matrices-and-determiminant -> SOLUTION: Determine the augmented matrix:p+s+c=600; 12p+10s+8c=5900; 0-s+c=175 Solve for the matrix and please show all work.       Log On

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Question 143429: Determine the augmented matrix:p+s+c=600; 12p+10s+8c=5900; 0-s+c=175
Solve for the matrix and please show all work.
Answer by Edwin McCravy(6939) About Me  (Show Source):
You can put this solution on YOUR website!
Determine the augmented matrix:p+s+c=600; 12p+10s+8c=5900; 0-s+c=175
Solve for the matrix and please show all work. 


++p%2B++s%2B+c+=+++600
12p%2B10s%2B8c+=+5900
0++-s%2B+c+=++175 

Solve for the matrix and please show all work. 

Put in all the 1 coefficients and make the 0 a 0p

1p%2B1s%2B1c=600
12p%2B10s%2B8c=5900
0p-1s%2B1c=175

Now erase all the letters and you have

drawing%28110%2C50%2C-10%2C10%2C-10%2C10%2C+line%283.8%2C10%2C3.8%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C1%2C1%2C1%2C600%2C12%2C10%2C8%2C5900%2C0%2C-1%2C1%2C175%29%29%0D%0A%29

The idea is to get three 0's in the lower left corner:
Multiply the top row through by -12

drawing%28200%2C50%2C-10%2C10%2C-10%2C10%2C+line%281.5%2C10%2C1.5%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C-12%2C-12%2C-12%2C-7200%2C12%2C10%2C8%2C5900%2C0%2C-1%2C1%2C175%29%29%0D%0A%29

Add the top row to the middle row:

drawing%28200%2C50%2C-10%2C10%2C-10%2C10%2C+line%281.5%2C10%2C1.5%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C-12%2C-12%2C-12%2C-7200%2C0%2C-2%2C-4%2C-1300%2C0%2C-1%2C1%2C175%29%29%0D%0A%29

Restore the top row

drawing%28200%2C50%2C-10%2C10%2C-10%2C10%2C+line%28-2%2C10%2C-2%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C1%2C1%2C1%2C600%2C0%2C-2%2C-4%2C-1300%2C0%2C-1%2C1%2C175%29%29%0D%0A%29

Multiply the bottom row by -2

drawing%28200%2C50%2C-10%2C10%2C-10%2C10%2C+line%28-2%2C10%2C-2%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C1%2C1%2C1%2C600%2C0%2C-2%2C-4%2C-1300%2C0%2C2%2C-2%2C-350%29%29%0D%0A%29
Add the middle row to the bottom row:

drawing%28200%2C50%2C-10%2C10%2C-10%2C10%2C+line%28-2%2C10%2C-2%2C-10%29%2C+%0D%0A%28matrix%283%2C4%2C1%2C1%2C1%2C600%2C0%2C-2%2C-4%2C-1300%2C0%2C0%2C-6%2C-1650%29%29%0D%0A%29

Now we've gotten three 0's in the bottom left.
So we make the matrix back into 3 equations:

1p%2B1s%2B1c=600
0p-2s-4c=-1300
0p%2B0s-6c=-1650

Drop the 0 terms and make the 1's invisible:

p%2Bs%2Bc=600
-2s-4c=-1300
-6c=-1650

Solve from the bottom to top.
Solve the bottom equation for c

-6c=-1650
%28-6c%29%2F%28-6%29=%28-1650%29%2F%28-6%29
c=275

Plug that into the second equation:

-2s-4c=-1300
-2s-4%28275%29=-1300
-2s-1100=-1300
-2s=-200
%28-2s%29%2F%28-2%29=%28-200%29%2F%28-2%29
s=100

Plug that and c=275 into the first
equation:

p%2Bs%2Bc=600
p%2B100%2B275=600
p%2B375=600
p=225

So

(p,s,c) = (225,100,275)

Edwin