Expert: Sherry Wallin - 11/30/2009

Question
Solve this system of equations:

200x+160y+60z=2100
400x+300y+200z=4500
x+y-2z=0

Answer
First things first, why not make these equations as simple as possible? You can divide out the GCF so for example in
200x+160y+60z=2100 each of the terms have a factor of 20 in them so get rid of it giving you a new equivalent equation: 10x + 8y + 3z = 105, likewise divide a 100 out of the 2nd equation giving you:
4x + 3y + 2z = 45 thus your new, equivalent system is:

10x + 8y + 3z = 105  call this eq 1
4x + 3y + 2z = 45   call this eq 2
 x +  y - 2z = 0    call this eq 3

multiply eq 3 by -4 getting -4x -4y + 8z = 0  call this eq 3a add it to eq 2 getting
-y + 10z = 45 call this eq 4

now multiply eq 3 by -10 getting -10x -10y +20 z = 0 add it to eq 1 getting
-2y +23z = 105  call this eq 5

Notice eq 4 and eq 5 only have y's and z's so eliminate one of those variables
multiply eq 4 by -2 getting 2y -20z = -90 add it to eq 5 getting
3z = 15 -> z = 5  back substitute into either of eq 4 or eq 5 z = 5 I choose eq 4 getting
-y + 10(5) = 45 -> y = 5

You are now armed with enough information to find x in any of the first three equations choose eq 3:
x + 5 -2(5) = 0 -> x = 5

Your proposed solution is the ordered triplet (5, 5, 5). If it is the solution it will make a true statement for all of the original equations


10x + 8y + 3z = 105  10(5)+8(5)+3(5) = 105
4x + 3y + 2z = 45   4(5)+3(5)+2(5) = 45
 x +  y - 2z = 0    5 + 5 -2(5) = 0

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