Question 1123039
a. 
{{{x-2y+3z = -7}}}
{{{2y = 6}}}
{{{2x-z = 5}}} 

Your matrix:

{{{matrix(4,3,1,	-2,	3,	-7,
0,	2,	0,	6,
2,	0,	-1,	5)}}}


Write down the main matrix and find its determinant:

{{{matrix(3,3,1,	-2,	3,	
0,	2,	0,	
2,	0,	-1)}}}

Δ = {{{-14}}}


Replace the 1st column of the main matrix with the solution vector and find its determinant:


{{{matrix(3,3, -7	,-2	,3,
6	,2	,0,
5	,0	,-1)}}}


Δ{{{1 = -28}}}

Replace the 2nd column of the main matrix with the solution vector and find its determinant:


{{{matrix(3,3,1	,-7,	3,
0	,6	,0,
2	,5	,-1)}}}


Δ{{{2 = -42}}}



Replace the 3rd column of the main matrix with the solution vector and find its determinant:

{{{matrix(3,3,
1	,-2,	-7,
0	,2	,6,
2,	0	,5)}}}


Δ{{{3 = 14}}}


{{{x[1]}}} = Δ{{{1}}} /Δ ={{{ (-28) / (-14) = 2}}}
{{{x[2]}}} = Δ{{{2}}} /Δ = {{{(-42) / (-14) = 3}}}
{{{x[3]}}} = Δ{{{3}}}/Δ ={{{ 14 / (-14) = -1}}}


Solution set:
{{{x[1] = 2}}}
{{{x[2] = 3}}}
{{{x[3]= -1}}}



b. 
{{{3x - y + 2z = 13}}}
 {{{ -x + 4y + 2z= -1}}}
{{{ 4y + 3z   = 4}}}


Your matrix

{{{matrix(4,3,
3	,-1	,2	,13,
-1	,4	,2	,-1,
0	,4	,3	,4)}}}


Write down the main matrix and find its determinant:



{{{matrix(3,3,
3	,-1	,2,
-1	,4	,2,
0	,4	,3)}}}


Δ = {{{1}}}


Replace the 1st column of the main matrix with the solution vector and find its determinant:


{{{matrix(3,3,13	,-1	,2,
-1,	4	,2,
4	,4	,3)}}}


Δ{{{1 = 1}}}


Replace the 2nd column of the main matrix with the solution vector and find its determinant:


{{{matrix(3,3,3,	13	,2,
-1	,-1	,2,
0	,4	,3)}}}

Δ{{{2 = -2}}}


Replace the 3rd column of the main matrix with the solution vector and find its determinant:


{{{matrix(3,3,3	,-1	,13,
-1	,4	,-1,
0	,4	,4)}}}


Δ{{{3 = 4}}}



{{{x[1]}}} = Δ{{{1}}} / Δ = {{{1 / 1 = 1}}}
{{{x[2]}}} = Δ{{{2}}} / Δ ={{{ (-2) / 1 = -2}}}
{{{x[3] }}}= Δ{{{3}}} / Δ ={{{ 4 / 1 = 4}}}



Solution set:

{{{x[1] = 1}}}
{{{x[2] = -2}}}
{{{x[3] = 4}}}