# For the given systems of linear equations, determine the values of b_1, b_2, text{ and } b_3 necessary for the system to be consistent. (Using matrices) x-y+3z=b_1 3x-3y+9z=b_2 -2x+2y-6z=b_3

For the given systems of linear equations, determine the values of necessary for the system to be consistent. (Using matrices)
$x-y+3z={b}_{1}$
$3x-3y+9z={b}_{2}$
$-2x+2y-6z={b}_{3}$
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Step 1
Consider
$x-y+3z={b}_{1}$
$3x-3y+az={b}_{2}$
$-2x+2y-6z={b}_{3}$

$\therefore \left[A|b\right]=\left[\begin{array}{ccccc}1& -1& 3& |& {b}_{1}\\ 3& -3& 9& |& {b}_{2}\\ -2& 2& -6& |& {b}_{3}\end{array}\right]$
${R}_{2}\to {R}_{2}-3{R}_{1},{R}_{3}\to {R}_{3}+2{R}_{1}$
$\left[A|b\right]\sim \left[\begin{array}{ccccc}1& -1& 3& |& {b}_{1}\\ 0& 0& 0& |& {b}_{2}-3{b}_{1}\\ 0& 0& 0& |& {b}_{3}+2{b}_{1}\end{array}\right]$
the Rank(A) will be equal to Rank(A|b) iff ${b}_{2}-3{b}_{1}=0$ and ${b}_{3}+2{b}_{1}=0$
which implies ${b}_{2}=3{b}_{1}$
and ${b}_{3}=-2{b}_{1}$
Hence , given system will be consistent iff $b=\left\{\left[\begin{array}{c}{b}_{1}\\ 3{b}_{1}\\ -2{b}_{1}\end{array}\right]|{b}_{1}\in \mathbb{R}\right\}$
Jeffrey Jordon