Review : Systems of Equations — Question 5

Question 5 – Solution

Strategy. Subtracting almost identical equations exposes the small coefficient that amplifies uncertainty.

Step 1: Solve and retain the excluded case. Subtract the first equation from the second to get $\varepsilon y=\delta$. For $\varepsilon>0$, $$\boxed{(x,y)=\left(2-\frac\delta\varepsilon, \frac\delta\varepsilon\right).}$$ If $\varepsilon=0$ and $\delta\ne 0$, there is no solution; if both are zero, the two equations coincide and every point on $x+y=2$ is a solution.

Step 2: Derive the exact uncertainty bound. The nominal solution is $(2,0)$. Each coordinate changes in magnitude by $|\delta|/\varepsilon$. Therefore $$\boxed{\max(|x-2|,|y|)\le\eta/\varepsilon.}$$ Equality occurs when $\delta=\eta$ or $\delta=-\eta$, so no smaller universal bound is possible.

Step 3: Quantify the nearly parallel example. For $\varepsilon=\eta=10^{-3}$, let $u=\delta/\varepsilon\in[-1,1]$. The full possible solution set is the segment $\boxed{(x,y)=(2-u,u),\ -1\le u\le 1}$, from $(3,-1)$ to $(1,1)$. The nominal answer satisfies the first equation exactly and has residual $-\delta$ in the second, of magnitude at most $0.001$. Nevertheless its coordinate error can be $1$. Small residuals do not always certify small solution errors. The graph uses true scales; the two measured lines are nearly indistinguishable, while the possible solution segment is visible.

Step 4: Give the exact design condition. The requested guarantee holds exactly when $\eta/\varepsilon\le 0.1$, or $\boxed{\varepsilon\ge 10\eta}$. Necessity follows by taking an endpoint error; sufficiency follows from the bound. Increasing $\varepsilon$ separates the line directions and reduces the effect of shifting one measurement’s right-hand side.