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

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

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

$$0 x+0 y=-7$$
$$0=-7$$ (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 $$y=r$$

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

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

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

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

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