Question 1135198

{{{5x - 6y + 4z = 15}}}.........eq.1
{{{7x + 4y - 3z = 19}}}.........eq.2
{{{2x + y + 6z = 46}}}.........eq.3
--------------------------------------------------


start with

{{{7x + 4y - 3z = 19}}}.........eq.2.......both sides multiply by {{{2}}}
{{{2x + y + 6z = 46}}}.........eq.3
----------------------------------------------
{{{14x + 8y - 6z = 38}}}.........eq.2
{{{2x + y + 6z = 46}}}.........eq.3
-------------------------------------------------add both

{{{14x + 8y - 6z +2x + y + 6z= 38+46}}}

{{{16x + 9y = 84}}}........solve for {{{x}}}

{{{16x = 84-9y}}}

{{{x = 84/16-9y/16}}}

{{{x = 21/4-9y/16}}}............eq.1a



go to


{{{5x - 6y + 4z = 15}}}.........eq.1
{{{2x + y + 6z = 46}}}.........eq.3......both sides multiply by {{{6}}}
--------------------------------------------------
{{{5x - 6y + 4z = 15}}}.........eq.1
{{{12x + 6y + 36z = 276}}}.........eq.3
---------------------------------------------------add both

{{{5x - 6y + 4z +12x + 6y + 36z = 15+276}}}

{{{17x  + 40z = 291}}}.......solve for {{{x}}}

{{{17x  = 291-40z }}}

{{{x  = 291/17-40z/17 }}}............eq.1b



from eq.1and eq.1b we have


{{{ 21/4-9y/16=291/17-40z/17}}}.......sole for {{{z}}}

{{{ 40z/17=291/17-21/4+9y/16}}}

{{{ 40z/17=9y/16 + 807/68}}}........both sides multiply by {{{17}}}

{{{ 40z=17(9y/16) +cross(17)( 807/cross(68)4)}}}

{{{ 40z=153y/16 + 807/4}}}

{{{ z= 153y/640 + 807/160}}}.............eq1c



go to {{{2x + y + 6z = 46}}}.........eq.3, substitute {{{x}}} from eq.1a and {{{z}}} from eq.1c


{{{2(21/4-9y/16) + y + 6(153y/640 + 807/160) = 46}}}....solve for {{{y}}}

{{{21/2 - 9y/8 + y +459y/320 + 2421/80 = 46}}}

{{{ - 9y/8 + y +459y/320 = 46-21/2- 2421/80 }}}

{{{ 419y/320 = 419/80}}}

{{{ y = (419/80)(320/419)}}}

{{{ y = (1/1)(4/1)}}}

{{{ y = 4}}}



go to {{{x = 21/4-9y/16}}}............eq.1a, substitute {{{y}}}

{{{x = 21/4-(9*4)/16}}}
{{{x = 21/4-9/4}}}
{{{x = 12/4}}}
{{{x = 3}}}



go to {{{ z=153y/640 + 807/160}}}.............eq1c, substitute {{{y}}}

{{{ z=(153*4)/640 + 807/160}}}

{{{ z=(153*cross(4))/(cross(640)160) + 807/160}}}

{{{ z=153/160 + 807/160}}}
{{{ z=960/160}}}
{{{ z=6}}}


so, your solutions are: {{{x = 3}}}, {{{y = 4}}},{{{ z = 6}}}