Solving a+ar+ar^2=7 and a^3+a^3r^3+a^3r^6=73

  • Thread starter santa
  • Start date
In summary, to find the values of 'a' and 'r' in the equations a+ar+ar^2=7 and a^3+a^3r^3+a^3r^6=73, you can use a trial-and-error method. One possible solution is a=1 and r=2, as shown in the provided link. Alternatively, you can sum the geometric equations and solve for 'a' and 'r'.
  • #1
santa
18
0
find a and r

[tex]a+ar+ar^2=7[/tex]

[tex]a^3+a^3r^3+a^3r^6=73[/tex]
 
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  • #2
santa said:
find a and r

[tex]a+ar+ar^2=7[/tex]

[tex]a^3+a^3r^3+a^3r^6=73[/tex]

http://www.imf.au.dk/kurser/algebra/E05/GBintro.pdf
 
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  • #3
OK but the solution where
 
  • #4
The first trial-and-error attempt I made turned out to be right: a=1, r=2,
1 + 2 + 4 = 7,
1 + 8 + 64 = 73.
 
  • #5
Not sure if there is more than one solution, but I found that a = 1 and r =2 works.
 
  • #6
a(1 + r + r^2) = 7

Then you can solve for 'a' or 'r', and substitute into the other equation. It's not pretty, but it will work!

I remember seeing a very similar problem in high school, must see if I can find it again.

In fact, you could try by summing geometric equations.

a + ar + ar^2 = a(1 - r^3)/(1 - r) = 7

a^3 + a^3.r^3 + a^3.r^6 = a^3(1 - r^9)/(1 - r^3) = 73

Divide the one sum by the other and see where it leads you.
 

1. What is the general approach to solving this system of equations?

The general approach to solving a system of equations is to isolate one variable in one equation and substitute it into the other equation. This will create a single equation with one variable, which can then be solved for that variable. The solution can then be substituted back into either of the original equations to find the other variable.

2. How do I determine which variable to isolate first?

In this system of equations, we can see that the first equation has a common factor of "a" in all three terms, while the second equation has a common factor of "a^3" in all three terms. Therefore, it would be most efficient to isolate "a" in the first equation and "a^3" in the second equation.

3. Can I use the substitution method to solve this system?

Yes, the substitution method can be used to solve this system of equations. As mentioned earlier, we can isolate one variable and substitute it into the other equation to create a single equation with one variable.

4. Is there a way to check my solution?

Yes, you can check your solution by substituting the values back into the original equations. If the solution is correct, it should satisfy both equations.

5. Are there any special cases for this system of equations?

Yes, there are two special cases for this system of equations. The first case is when r = 1, which would make both equations equivalent to a + a + a = 7 and a^3 + a^3 + a^3 = 73. In this case, the solution would be a = 7/3 and r = 1. The second case is when r = -1, which would make the first equation equivalent to a - a + a = 7 and the second equation equivalent to a^3 - a^3 + a^3 = 73. In this case, the solution would be a = 7 and r = -1.

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