Question 1090977
Solve the system for {{{x}}},{{{y}}},{{{z}}}
{{{2x+3y-z=4}}}.....eq.1
{{{3x-y+2z=5}}}.....eq.2
{{{x-4y+3z=1}}}.....eq.3

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{{{2x+3y-z=4}}}.....eq.1...solve for {{{z}}}

{{{z=2x+3y-4}}}.....eq.(1z)

{{{3x-y+2z=5}}}.....eq.2..solve for {{{z}}}

{{{2z=-3x+y+5}}}

{{{z=-3x/2+y/2+5/2}}}....eq.(2z)

{{{x-4y+3z=1}}}.....eq.3..........solve for {{{z}}}

{{{3z=-x+4y1}}}

{{{z = -x/3 + (4 y)/3 + 1/3}}}....eq.(3z)


from eq.(1z)  and eq.(2z):

{{{2x+3y-4=-3x/2+y/2+5/2}}}...solve for {{{x}}}

{{{4x+6y-8=-3x+y+5}}}
{{{4x+3x=y-6y+5+8}}}
{{{7x=-5y+5+8}}}
{{{x=-5y/7+13/7}}}.....(a)

from eq.(2z)  and eq.(3z):

{{{-3x/2+y/2+5/2=-x/3 + (4 y)/3 + 1/3}}}....both sides multiply by {{{6}}}

{{{-9x+3y+15=-2x + 8 y + 2}}}
{{{-8y+3y+15-2=-2x + 9x}}}
{{{-5y+13=7x}}}
{{{x=-5y/7+13/7}}}.....(b)

{{{3x-y+2z=5}}}.....eq.2....substitute {{{x}}} and {{{z}}}

{{{3(-5y/7+13/7)-y+2(-3x/2+y/2+5/2)=5}}}.....eq.2

{{{highlight(y = 13/5 - (7 x)/5)}}}-> real solution

{{{z=2x+3y-4}}}.....eq.(1z)......substitute {{{y}}}

{{{z=2x+3(13/5 - (7x)/5)-4}}}

{{{highlight(z  = 19/5 - (11x)/5)}}}-> real solution

{{{highlight(x = 5n + 4)}}}, {{{highlight(y = -7n - 3)}}}, {{{highlight(z = -11n - 5)}}}, where {{{n}}} element {{{Z}}}

integer solutions:

({{{x}}},{{{y}}},{{{z}}})=({{{5n + 4}}}, {{{-7n - 3}}}, {{{-11n - 5}}})

find some real take real solutions and equal to zero:

{{{ y=13/5 - (7 x)/5}}}

{{{ 13/5 - (7 x)/5=0}}}...solve for {{{x}}}
{{{ 13 - 7x=0}}}
{{{ 13 =7x}}}
{{{highlight(x=13/7)}}}

plug in

{{{z=19/5 - (11x)/5)}}}

{{{z=19/5 - (11(13/7))/5)}}}

{{{highlight(z=-2/7)}}}

find {{{y}}}:

{{{y = 13/5 - (7(13/7))/5}}}

{{{highlight(y = 0)}}}


real solutions:({{{x}}},{{{y}}},{{{z}}})=({{{13/7}}},{{{0}}},{{{-2/7}}})


or
if {{{n=0}}}
({{{x}}},{{{y}}},{{{z}}})=({{{ 4}}},{{{ - 3}}},{{{ - 5}}})
if {{{n=1}}}
({{{x}}},{{{y}}},{{{z}}})=({{{9}}}, {{{-10}}}, {{{-16}}}) ...and so on