Question

The following set of equations has

3x + 2y + z = 4

x - y + z = 2

-2x + 2z = 5

• A. No solution
• B. A unique solution
• C. Multiple solutions
• D. An inconsistency

Answer 1

The correct option is B A unique solution

Using $R_1\leftrightarrow R_2$

[A : B] = $\left[\begin{array}{ccccc}1& -1& 1& :& 2\\ 3& 2& 1& :& 4\\ -2& 0& 2& :& 5\end{array}\right]$

$R_1\leftrightarrow R_2-3R_1$

$R_3\leftrightarrow R_3-2R_2$

- $\left[\begin{array}{ccccc}1& -1& 1& :& 2\\ 0& 5& -2& :& -2\\ 0& -2& 4& :& 9\end{array}\right]$

$R_3\to R_3+\frac{2}{5}{R}_2$

- $\left[\begin{array}{ccccc}1& -1& 1& :& 2\\ 0& 5& -2& :& -2\\ 0& -2& \frac{16}{5}& :& \frac{41}{5}\end{array}\right]$

$\rho \left(A\right)$ = $\rho$l [A : B] = 3 = Number of unknowns

\(\\therefore ) Unique solution.