Review : Systems of Equations — Question 4

Question 4 – Solution

Strategy. Redundant measurements can reveal a contradiction without identifying its source.

Step 1: Derive the consistency condition and solution. Adding the three pairwise equations gives $2(x+y+z)=a+b+c$. Thus consistency requires $\boxed{a+b+c=2d}$. The pairwise equations alone uniquely give $$\boxed{x=\frac{a+c-b}{2},\quad y=\frac{a+b-c}{2},\quad z=\frac{b+c-a}{2}.}$$ These values satisfy all three pairwise equations, and their total is $(a+b+c)/2$. Therefore the displayed condition is also sufficient.

Step 2: Diagnose the recorded discrepancy. For $(a,b,c)=(2,3,4)$, the unique triple is $(3/2,1/2,5/2)$. Its total is $9/2$, not $5$. Hence the full recorded system has no solution; rounding the computed triple would not repair an exact contradiction.

Step 3: Enumerate every single-reading correction. The exact total requires the sum of the pairwise readings to be $10$ rather than $9$. If exactly one pairwise reading changes, its correction must be $+1$. There are precisely three possibilities: $$\begin{array}{c|c|c} \text{Changed reading}&(a,b,c)&(x,y,z)\\\hline a:2\to 3&(3,3,4)&(2,1,2)\\ b:3\to 4&(2,4,4)&(1,1,3)\\ c:4\to 5&(2,3,5)&(2,0,3) \end{array}$$ Each triple satisfies its corrected pairwise readings and has total $5$.

Step 4: State the identification limit. All three corrected triples are nonnegative. Thus even with nonnegativity, the measurements cannot select the faulty reading. The redundancy detects that the four recorded equations cannot all be true. Locating the error requires another independent measurement or an additional justified restriction; it does not follow merely from detecting the inconsistency.