The fourth column sums to 45. The fifth column sums to 60. Here’s one way to see this:
The three rows represent three equations:
- 2A + B + C + D = 80
- A + 2B + C + D = 95
- 3A + 2B= 80
Subtracting equation (1) from equation (2) yields: - A + B = 15, or:
B = A + 15.
Substituting this into the third equation yields 3A + 2(A + 15) = 80, that is, 5A + 30 = 80, giving:
A = 10
and so
B = 25.
The value of the fifth column is thus A + 2B = 10 + 50 = 60.
Returning to the original three equations, now subtract equation (3) from equation (2). This gives: -2A + C + D = 15, that is,
C + D = 2A + 15 = 35.
Consequently, the sum of values in the fourth column is A + C + D = 10 + 35 = 45.
This approach to the problem, although swift, feels ad hoc. It is not clear to the reader how I, the author, came upon this approach – and, to be honest, it is not clear to me how I, the author, could describe a general approach of this style that would always solve problems of this type. All I can say is that I “played” with the equations for a while, using my experience with problems like these to hone in on a quick approach to its solution. This is not very illuminating!
What is needed is a systematic set of instructions that anyone could follow (better yet, that a computer could follow) to solve, or at least analyze, systems of equations. In the early 1800s the great German mathematician and physicist Carl Friedrich Gauss developed just such a method! For more on this, click here.
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