Question

Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13

Answer 1

The given system of equations can be expressed as follows:
\[AX = B \]
Here, 
\[ A = \begin{bmatrix}2 & 5 \\ 6 & 15\end{bmatrix}, X = \binom{x}{y}\text{ and }B = \binom{7}{13}\]
Now, 
\[\left| A \right| = \begin{vmatrix}2 & 5 \\ 6 & 15\end{vmatrix}\]
\[ = \left( 30 - 30 \right)\]
\[ = 0\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A =\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = - 1^{1 + 1} \left( 15 \right) = 15, C_{12} = - 1^{1 + 2} \left( 6 \right) = - 6\]
\[ C_{21} = - 1^{2 + 1} \left( 5 \right) = - 5, C_{22} = - 1^{2 + 2} \left( 2 \right) = 2\]
\[adj A = \begin{bmatrix}15 & - 6 \\ - 5 & 2\end{bmatrix}^T \]
\[ = \begin{bmatrix}15 & - 5 \\ - 6 & 2\end{bmatrix}\]
\[\left( adj A \right) B = \begin{bmatrix}15 & - 5 \\ - 6 & 2\end{bmatrix}\binom{7}{13}\]
\[ = \binom{105 - 65}{ - 42 + 26}\]
\[ = \binom{40}{ - 16} \neq 0\]
Hence, the given system of equations is inconsistent.