Consider the system x1 x2 1 x1 ex2 2. When e is close to 1, the lines that make up the
Chapter 8, Problem 12(choose chapter or problem)
Consider the system
\(\begin{array}{c} x_{1}+x_{2}=1 \\ x_{1}+\varepsilon x_{2}=2 \end{array}\)
When \(\epsilon\) is close to 1, the lines that make up the system are almost parallel.
(a) Use Cramer’s rule to show that a solution of the system is
\(x_{1}=1-\frac{1}{\varepsilon-1}, \quad x_{2}=\frac{1}{\varepsilon-1}\).
(b) The system is said to be ill-conditioned since small changes in the input data (for example, the coefficients) causes a significant or large change in the output or solution. Verify this by finding the solution of the system for \(\epsilon =1.01\) and then for \(\epsilon =0.99\).
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