System of equations

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Riddame

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Can someone help me answer this, it's driving me insane. Its a system of equations and i need to know, x,y,z and w as functions of only a,b,c and d (which are constants)
x=(0.564y+0.564z+0.564w)/(1-0.564),
y=(0.385x+0.385z+0.385w)/(1-0.385),
z=(0.031x+0.031y+0.031w)/(1-0.031),
w=(0.02x+0.02y+0.02z)/(1-0.02),
480=ax+by+cz+dw
 
Riddame said:
Can someone help me answer this, it's driving me insane. Its a system of equations and i need to know, x,y,z and w as functions of only a,b,c and d (which are constants)
x=(0.564y+0.564z+0.564w)/(1-0.564),
y=(0.385x+0.385z+0.385w)/(1-0.385),
z=(0.031x+0.031y+0.031w)/(1-0.031),
w=(0.02x+0.02y+0.02z)/(1-0.02),
480=ax+by+cz+dw
Click to expand...
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Please share your work/thoughts about this assignment.
 
Riddame said:
Can someone help me answer this, it's driving me insane. Its a system of equations and i need to know, x,y,z and w as functions of only a,b,c and d (which are constants)
x=(0.564y+0.564z+0.564w)/(1-0.564),
y=(0.385x+0.385z+0.385w)/(1-0.385),
z=(0.031x+0.031y+0.031w)/(1-0.031),
w=(0.02x+0.02y+0.02z)/(1-0.02),
480=ax+by+cz+dw
Click to expand...
I'd start by putting each equation in the form _x + _y + _z + _w = _ , and hope to solve the traditional way. But on the way to that, you may see something special about the equations, involving x+y+z+w, which you might give a name (let's call it u). This will provide a shortcut you could take, by expressing the variables in terms of u.

But the more work you show, the better we can help you take your own ideas and make them work.
 
I am sorry I didn't explain in better details, so here is the whole problem:
This is not an exercise or homework, this is a real problem that I am trying to solve for the company I work for.
In this particular problem, we provide 4 types of services to our clients.
Service X
Service Y
Service Z
Service W
I want to know how many of each service we do daily, this is the information I have:
Time it takes to do service X = a
Time it takes to do service Y = b
Time it takes to do service Z = c
Time it takes to do service W = d
(a,b,c and d are all constants that I know)
I also know the proportion of each service in relation to the sum of all the service done daily:
Service X represents 56,4% of all services
Service Y represents 38,5% of all services
Service Z represents 3,1% of all services
Service W represents 2,0% of all services
Furthermore "e" is the time of the shift

The equations above are where I stopped. I tried solving in Wolfram Alpha but I do not know enough about the program to type these equations.
Basically I need functions as follows:
X=f(a,b,c,d,e)
Y=f(a,b,c,d,e)
Z=f(a,b,c,d,e)
W=f(a,b,c,d,e)
I need them as functions of the time of each service because i am trying to implement this in an excel spreadshet where the user can change the times.
 
If you follow my suggestions, it isn't hard; but I don't have my notes, or time to do the work for you, at the moment, so I'll try to get back to it when I can, if someone else hasn't.
 
Riddame said:
Can someone help me answer this, it's driving me insane. Its a system of equations and i need to know, x,y,z and w as functions of only a,b,c and d (which are constants)
x=(0.564y+0.564z+0.564w)/(1-0.564),
y=(0.385x+0.385z+0.385w)/(1-0.385),
z=(0.031x+0.031y+0.031w)/(1-0.031),
w=(0.02x+0.02y+0.02z)/(1-0.02),
480=ax+by+cz+dw
Click to expand...
x=(0.564y+0.564z+0.564w)/(1-0.564) ....... \(\displaystyle \to \\\\\) x = 0.564*(y + z + w)/0.436....... \(\displaystyle \to \\\\\) 0.7730 * x = y + z + w ......(1)

y=(0.385x+0.385z+0.385w)/(1-0.385)....... \(\displaystyle \to \\\\\) y = 0.385*(x + z + w)/0.615....... \(\displaystyle \to \\\\\) 1.5974 * y = x + z + w .....(2)

z=(0.031x+0.031y+0.031w)/(1-0.031)....... \(\displaystyle \to \\\\\) z = 0.031*(y + z + x)/0.969....... \(\displaystyle \to \\\\\) 31.2581 * z = y + w + x.....(3)

w=(0.02x+0.02y+0.02z)/(1-0.02)................ \(\displaystyle \to \\\\\) w = 0.02*(y + z + x)/0.98....... \(\displaystyle \to \\\\\) 49 * w = y + z + x.................(4)

480=ax+by+cz+dw

What do you get if you add (1), (2), (3) & (4)?

Now follow suggestions in response #3
 
My alternative approach was to rewrite the equations (by multiplying by the unsimplified denominators and distributing) as:

x = 0.564(x+y+z+w)​
y = 0.385(x+y=z+w)​
z = 0.031(x+y+z+w)​
w = 0.02(x+y+z+w)​

so that if u = x+y+z+w, we can replace each variable in the last equation with x = 0.564u, and so on, leaving an equation that can be solved for u, and thence for x, y, z, w, in terms of a, b, c, d.
 
Thank you everyone! With the tips that you gave me i was able to solve it
 
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