Question 1205351

theater has{{{ 700}}} ​seats

let orchestra seats be {{{x}}}, main seats be {{{y}}} and balcony seats{{{ z}}}

{{{ x+y+z=700}}}....eq.1

 if you pay ${{{50 }}}for orchestra seat, main seats for ${{{40}}}, and balcony seats for ${{{25}}}, and if all seats are sold the gross revenue is ${{{26250}}}, we have 

{{{50x+40y+25z= 26250}}}.....eq.2

If all the main and balcony seats are​ sold, but only {{{half }}}the orchestra seats are​ sold, the gross revenue is ${{{22750}}}

{{{50(x/2)+40y+25z= 22750}}}.....eq.3


solve the system:

{{{ x+y+z=700}}}....eq.1
{{{50x+40y+25z= 26250}}}.....eq.2
{{{50(x/2)+40y+25z= 22750}}}.....eq.3....simplify
-------------------------------------------------------
{{{ x+y+z=700}}}....eq.1
{{{50x+40y+25z= 26250}}}.....eq.2
{{{25x+40y+25z= 22750}}}.....eq.3
-------------------------------------------------------


start with

{{{ x+y+z=700}}}....eq.1.....multiply by {{{50}}}
{{{50x+40y+25z= 26250}}}.....eq.2
-------------------------------------------

{{{ 50x+50y+50z=35000}}}....eq.1
{{{50x+40y+25z= 26250}}}.....eq.2
-------------------------------------------------subtract

{{{ 50x-50x+50y-40y+50z-25z=35000-26250}}}
{{{ 10y+25z=8750}}}....solve for {{{y}}}
{{{ 10y=8750-25z}}}
{{{ y=(8750-25z)/10}}}
{{{ y=875-2.5z}}}..........eq.1a



{{{ 50x+40y+25z= 26250}}}....eq.2
{{{25x+40y+25z= 22750}}}.....eq.3....multiply by {{{2}}}
-------------------------------------------------------

{{{ 50x+40y+25z= 26250}}}....eq.2
{{{50x+80y+50z= 45500}}}.....eq.3
------------------------------------------------subtract

{{{ 50x+80y+50z-(50x+40y+25z) =45500-26250}}}

{{{ 40y + 25z =19250}}}....solve for {{{y}}}
{{{ 40y =19250-25z}}}
{{{ y =19250/40-25z/40}}}
{{{ y =1925/4 - (5z)/8}}}

 {{{ y =481.25-0.625z}}}.......eq.1b


from eq.1a and eq.1b we have


{{{ 875-2.5z=481.25-0.625z}}}

{{{ 875-481.25=2.5z-0.625z}}}

{{{ 393.75=1.875 z}}}

{{{ z=393.75/1.875 }}}

{{{ z=210}}}


go to 


{{{ y =481.25-0.625z}}}.......eq.1b, substitute {{{z}}}

{{{ y =481.25-0.625*210}}}

{{{ y =350}}}


go to


{{{ x+y+z=700}}}....eq.1, substitute {{{y}}} and {{{z}}}
{{{ x+350+210=700}}}
{{{ x+560=700}}}
{{{ x=700-560}}}
{{{ x=140}}}


there is {{{140}}} orchestra seats, {{{350}}} main seats, and {{{ 210}}} balcony seats