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$(30$ points$)$ Consider the system of following linear equations

$2\mathrm{.}5x_1-3\mathrm{.}4x_2+17\mathrm{.}4x_3-6\mathrm{.}7x_4+10x_5=-3$

$7x_1-3\mathrm{.}2x_2+19\mathrm{.}4x_3+5\mathrm{.}7x_4+13x_5=-1$

$2x_1+14\mathrm{.}4x_3+11x_5=5$

$6\mathrm{.}9x_1-5\mathrm{.}1x_2+4\mathrm{.}6x_3+3x_4+x_5=3$

$-9\mathrm{.}2x_2+19x_3+5x_4+x_5=-7$