SOLUTION: Solve these equations simultaneously to find a and r: a(r^8 - 8r^5)=0 a(1-r^5)/(1-r) = 93

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Question 1035273: Solve these equations simultaneously to find a and r:
a(r^8 - 8r^5)=0
a(1-r^5)/(1-r) = 93
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
system%28a%28r%5E8+-+8r%5E5%29=0%2C+a%281-r%5E5%29%2F%281-r%29+=+93%29 

Factor out r³ in the first equation
Multiply both sides of second equation by (1-r).
[Note that r cannot equal to 1 for that would make
the denominator of the original second equation 0.

system%28a%28r%5E3%5E%22%22%28r%5E5-8%29%29=0%2C+a%281-r%5E5%29=93%281-r%29%29  

Write the first equation as

ar%5E3%28r%5E5-8%29=0

Use the zero-factor principle:

 

Now we must find corresponding values
for "a":

Substitute a=0 in the second equation

a%281-r%5E5%29=93%281-r%29
0%281-r%5E5%29=93%281-r%29 
0=93%281-r%29
0=1-r
r=1

But we must disregard that value since it
causes the denominator in the original second
equation to become 0.  Therefore we must
also disregard a=0.

Substitute r=0 in the second equation

a%281-r%5E5%29=93%281-r%29
a%281-0%5E5%29=93%281-0%29 
a=93

So one solution is (a,r) = (93,0)

Substitute r=root%285%2C8%29 in the second equation

a%281-r%5E5%29=93%281-r%29
a%281-%28root%285%2C8%29%29%5E5%29=93%281-%28root%285%2C8%29%29%5E%22%22%29 
a%281-8%29=93%281-root%285%2C8%29%29
-7a=93%281-root%285%2C8%29%29
a=expr%28-93%2F7%29%281-root%285%2C8%29%29

Although that satisfies the second equation, we must check to
see if it satisfies the first equation as well:

a%28r%5E8+-+8r%5E5%29=0






The fractional powers of 8 are irrational, so none
of those factors can equal 0, so the equation cannot be true
so r=root%285%2C8%29 does not produce a solution.
However we could not have known this without substituting
in the first equation.

There is but one solution: (a,r) = (93,0)

Edwin