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Question 1101027: Parcel A and B together weigh 24 kgs, but parcel B and C together weighs only 21 kgs. When parcel A and C are paired, the combined weight is 27kgs. If the weight of parcel A is represented by a, parcel B by b and parcel C by c; derive a system of three equations for a, b and c. Using the inverse method derive the combined weight of parcel A, B and C.
Found 2 solutions by ikleyn, richwmiller:
Answer by ikleyn(52799)   (Show Source):
You can put this solution on YOUR website! .
a + b = 24, (1)
b + c = 21, (2)
a + c = 27. (3)
Add all the three eqs (1), (2) and (3) (both sides).
You will get
2a + 2b + 2c = 24+21+27, or
2(a + b + c) = 72, which implies
a + b + c = 36. (4)
Now subtract eq(1) from eq(4) (both sides). You will get c = 36-24 = 12.
Next subtract eq(2) from eq(4) (both sides). You will get a = 36-21 = 15.
Finally, subtract eq(3) from eq(4) (both sides). You will get b = 36-27 = 9.
It is THE STANDARD METHOD for solving problems like this.
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The problem asks: "Using the inverse method derive the combined weight of parcel A, B and C."
For me, this formulation, if to interpret each word literally, makes no sense.
But in any case, in my previous solution above I derived the expression
a + b + c = 36 (equation (4) ).
I didn't use any inverse method for it.
I simply added the three equations (1),(2) and (3), then divided the result by 2.
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The method of solvinng such problems was described in my lesson
- The tricks to solve some word problems with three and more unknowns using mental math
in this site.
Answer by richwmiller(17219)  (Show Source):
You can put this solution on YOUR website! original
a + b = 24
b + c = 21
a + c = 27
matrix
1 1 0 24
0 1 1 21
1 0 1 27
determinant =2
inverse matrix fractional form
1/2 -1/2 1/2
1/2 1/2 -1/2
-1/2 1/2 1/2
24 21 27
solutions and sum
15 + 9 + 12 =36
The process is much too long to present here.
I suppose that the teacher is introducing inverse matrix method and started with an easy system of equations. I can only assume that such a problem was given to practice the inverse matrix method so that you could check your results easily with Ikleyn's method.
Ikleyn is correct in saying that her method is THE STANDARD METHOD for solving problems like this.
There are several inverse matrix methods including using gauss jordan, identity matrix and using determinants, cofactors and and adjoint.
I used determinant, cofactor and adjoint.
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