SOLUTION: Solve for two equations 100x+95y+10z= 54(x+y+z) 95x+10y+5z= 33(x+y+z) How to find x,y,z? Answer should be: x=3, y=2, z=5

Algebra ->  Finance -> SOLUTION: Solve for two equations 100x+95y+10z= 54(x+y+z) 95x+10y+5z= 33(x+y+z) How to find x,y,z? Answer should be: x=3, y=2, z=5      Log On


   



Question 1157953: Solve for two equations
100x+95y+10z= 54(x+y+z)
95x+10y+5z= 33(x+y+z)
How to find x,y,z?
Answer should be: x=3, y=2, z=5

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
100x%2B95y%2B10z=+54%28x%2By%2Bz%29
100x%2B95y%2B10z=+54x%2B54y%2B54z
100x%2B95y%2B10z-54x-54y-54z=0
46x%2B41y-44z=0........solve for z
46x%2B41y=44z
z=46x%2F44%2B41y%2F44
z=23x%2F22%2B41y%2F44...........eq.1


95x%2B10y%2B5z=+33%28x%2By%2Bz%29
95x%2B10y%2B5z=+33x%2B33y%2B33z
95x%2B10y%2B5z-33x-33y-33z=0
62x+-+23y+-+28z+=+0........solve for z
62x+-+23y+=28z+
z=62x%2F28+-+23y%2F28
z=31x%2F14+-+23y%2F28...........eq.2


from eq.1 and eq.2 we have

23x%2F22%2B41y%2F44=31x%2F14+-+23y%2F28.......solve for y
23y%2F28%2B41y%2F44=31x%2F14+-+23x%2F22
23y%2F28%2B41y%2F44=31x%2F14+-+23x%2F22
%28135y%29%2F77=%2890+x%29%2F77
135y=77%2890+x%29%2F77
135y=90x
y=90x%2F135
y=+2x%2F3.............1a



go to z=31x%2F14+-+23y%2F28...........eq.2, substitute y

z=31x%2F14+-+%2823%282x%2F3%29%29%2F28
z=31x%2F14+-+%2846x%2F3%29%2F28
z=31x%2F14+-+23x%2F42
z=5x%2F3.............1b


Real solution:
y=+2x%2F3
z=5x%2F3

Integer solution:
y=+2x
z=5x+

=>since x could be any number 3n (because of fraction in real solution), integer solution is

x+=+3n, y+=+2n, z+=+5n, where n element Z

if we start with n=1, solution is highlight%28x+=+3%29, highlight%28y+=+2%29, highlight%28z+=+5%29

check if statement is true if x+=+3, y+=+2, z+=+5 :

100x%2B95y%2B10z=+54%28x%2By%2Bz%29
100%2A3%2B95%2A2%2B10%2A5=+54%283%2B2%2B5%29
300%2B190%2B50=+54%2810%29
540=+540=>true

let's check if n=2, solution is x+=+6, y+=+4, z+=+10
100x%2B95y%2B10z=+54%28x%2By%2Bz%29
100%2A6%2B95%2A4%2B10%2A10=+54%286%2B4%2B10%29
600%2B380%2B100=+54%2820%29
1080=+1080=>true