Question

Show that each one of the following systems of linear equation is inconsistent:
2x + 3y = 5
6x + 9y = 10

Answer 1

The given system of equations can be expressed as follows:
\[AX = B \]
Here,
\[ A = \begin{bmatrix}2 & 3 \\ 6 & 9\end{bmatrix}, X = \binom{x}{y}\text{ and }B = \binom{5}{10}\]
Now, 
\[\left| A \right| = \begin{vmatrix}2 & 3 \\ 6 & 9\end{vmatrix}\]
\[ = \left( 18 - 18 \right)\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} { \text{ be the cofactors of the elements a }}_{ij}\text{ in }A =\left[ a_{ij} \right].\text{ Then, }\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \left( 9 \right) = 9 , C_{12} = \left( - 1 \right)^{1 + 2} \left( 6 \right) = - 6 \]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \left( 3 \right) = - 3, C_{22} = \left( - 1 \right)^{2 + 2} \left( 2 \right) = 2\]
\[adj A = \begin{bmatrix}9 & - 6 \\ - 3 & 2\end{bmatrix}^T \]
\[ = \begin{bmatrix}9 & - 3 \\ - 6 & 2\end{bmatrix}\]
\[\left( adj A \right) B = \begin{bmatrix}9 & - 3 \\ - 6 & 2\end{bmatrix}\binom{5}{10}\]
\[ = \binom{45 - 30}{ - 30 + 20}\]
\[ = \binom{15}{ - 10} \neq 0\]
Hence, the given system of equations is inconsistent.