Question 106811
Elimination method is perfect for this system!


First let's eliminate x!  To do that, let's use {{{x+y+z=30000}}} and {{{.02x+.03y+.06z=1110 }}}.

Let's multiply {{{x+y+z=30000}}} by 2!


{{{2x+2y+2z=60000}}}

Now, let's multiply {{{0.02x+0.03y+0.06z=1110 }}} by 100!

{{{2x+3y+6z=111000}}}


It's elimination time!


Line up the equations

_____________________________2x+3y+6z=111000
_____________________________2x+2y+2z=60000


If you subtract 
{{{2x+2y+2z=60000}}} from {{{2x+3y+6z=111000}}},

you'll get 

{{{y+4z=51000}}}

Solving {{{y+4z=51000}}}.

{{{y=-4z+51000}}}


Now, let's eliminate y! we'll do that by using {{{0.02x+0.03y+0.06z=1110 }}} and   
{{{0.02x+150=0.03y}}}

Solving {{{0.02x+150=0.03y}}},

{{{0.02x-0.03y=-150}}}

Multiplying this by 100,

{{{2x-3y=-15000}}}

Multiplying {{{.02x+.03y+.06z=1110 }}} by 100,


{{{2x+3y+6z=111000}}}


it's elimination time!


Line up the equations!


______________________________2x+3y+6z=111000
______________________________2x-3y   =-15000

If you add these, you'll get:

{{{4x+6z=96000}}}

Solving this,

{{{4x=-6z+96000}}}
{{{x=-3z/2+24000}}}

Substituting {{{x=-3z/2+24000}}} and {{{y=-4z+51000}}} to {{{2x-3y=-15000}}},

{{{2(-3z/2+24000)-3(-4z+51000)=-15000}}}
{{{-3z+48000-(-12z+153000)=-15000}}}
{{{-3z+48000+12z-153000=-15000}}}
{{{(-3z+12z)+(48000-153000)=-15000}}}
{{{9z-105000=-15000}}}
{{{9z=-15000+105000}}}
{{{9z=90000}}}
{{{z=10000}}}

Yahoo! we found the value of z at last!

But wait, there's more!

Remember that {{{x=-3z/2+24000}}}. So,

{{{x=-3(10000)/2+24000}}}
{{{x=-30000/2+24000}}}
{{{x=-15000+24000}}}
{{{x=13000}}}


Also, 
{{{y=-4z+51000}}}. So,

{{{y=-4(10000)+51000}}}
{{{y=-40000+51000=11000}}}


therefore, x=13000, y=11000, and z=10000.


Power up,
HyperBrain!