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• Notes

$x-y=5$
$2 x-2 y=3$

$\left(\begin{array}{cc|c}{1} & {-1} & {5} \\ {2} & {-2} & {3}\end{array}\right) R_{2} \rightarrow R_{2}- 2 R_{1}$

$\left(\begin{array}{cc|c}{1} & {-1} & {5} \\ {0} & {0} & {-7}\end{array}\right)$

$0 x+0 y=-7$
$0=-7$ (false equation)

The system has No sol.

$x-y=5$
$2 x-2 y=10$

$\left(\begin{array}{cc|c}{1} & {-1} & {5} \\ {2} & {-2} & {10}\end{array}\right) R_{2}\rightarrow R_{2} - 2 R_{1}$

$\left(\begin{array}{cc|c}{1} & {-1} & {5} \\ {0} & {0} & {0}\end{array}\right) \Rightarrow x-y=5$

$0=0$

No. of var. $>$ No. of eq.

Infinit No. of sol.

No. of free var. = No. of var. - No. of eq.

$=2-1$

$=1$ free var.

Let $y=r$

$x-y=5$
$x-r=5$
$x=5+r$

$x=5+r$
$y=r$

$\left(\begin{array}{c}{x} \\ {y}\end{array}\right)=\left(\begin{array}{c}{5+r} \\ {r}\end{array}\right)$

$\left(\begin{array}{l}{x} \\ {y}\end{array}\right)=r\left(\begin{array}{l}{1} \\ {1}\end{array}\right)+\left(\begin{array}{l}{5} \\ {0}\end{array}\right)$