Question

For the system of equations:
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
(a) there is only one solution
(b) there exists infinitely many solution
(c) there is no solution
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{a}\right)\mathrm{there}\mathrm{is}\mathrm{only}\mathrm{one}\mathrm{solution} \\ \text{The given system of equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 4\end{array}\right] \\ \text{Here,} \\ \mathit{\text{A}}\text{=}\left[\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right]\text{,}\mathit{\text{X}}\text{=}\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\text{and}\mathit{\text{B}}=\left[\begin{array}{c}1\\ 2\\ 4\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\\ 5& 5& 9\end{array}\right| \\ =1\left(9-15\right)-2\left(18-15\right)+3\left(10-5\right) \\ =-6-6+15 \\ =3\ne 0 \\ \Rightarrow \left|A\right|\text{≠ 0} \\ \text{So, the given system of equations has a unique solution.} \end{gathered}$$