Show that each one of the following systems of equations is inconsistent, $x+2 y=9$ $2 x+4 y=7$

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Show that each one of the following systems of equations is inconsistent,
$x+2 y=9$
$2 x+4 y=7$

Show that each one of the following systems of equations is inconsistent,
$x+2 y=9$
$2 x+4 y=7$

Solution:

We need to prove: Set of given lines are inconsistent.
The given set of lines are: –
$\begin{array}{l} x+2 y=9 \\ 2 x+4 y=7 \end{array}$
Now, converting the following equations in matrix form,
$\begin{array}{l} \mathrm{AX}=\mathrm{B} \\ \left(\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} 9 \\ 7 \end{array}\right) \\ \mathrm{R}_{2}-2 \mathrm{R}_{1} \\ \left(\begin{array}{ll} 1 & 2 \\ 0 & 0 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{c} 9 \\ -11 \end{array}\right) \end{array}$
Again on converting into equation form, we obtain
$\begin{array}{l} x+2 y=9 \\ 0 x+0 y=-11 \\ \therefore 0=-11 \end{array}$
Which is not true
$\therefore \mathrm{x}+2 \mathrm{y}=9$
$2 \mathrm{x}+4 \mathrm{y}=7$ are inconsistent