17. Determine all real values of p and r which satisfy the following system of equations:
p+pr+pr^2=26 ,p^2r+p^2r^2+p^2r^3=156
Any tips help, but please show your work.
p+pr+pr^2=26
p(1+r+r^2) = 26
26 = 2*13
2(1+3+9) = 26
p=2, r=3
4*3(1+3+9) = 156
There is also the solution that
2*9(1 + 1/3 + 1/9) = 26
so p = 18 and q = 1/3
To determine the values of p and r that satisfy the given system of equations, we will solve the equations step by step. Let's begin:
Step 1: Simplify the first equation:
p + pr + pr^2 = 26
Step 2: Rewrite the first equation in terms of p:
p(1 + r + r^2) = 26
p = 26 / (1 + r + r^2) ---> Equation (1)
Step 3: Substitute the value of p from Equation (1) into the second equation:
(p^2)r + (p^2)(r^2) + (p^2)(r^3) = 156
Replace p with the expression obtained in Equation (1):
[(26 / (1 + r + r^2))^2]r + [(26 / (1 + r + r^2))^2](r^2) + [(26 / (1 + r + r^2))^2](r^3) = 156
Step 4: Simplify the above equation:
[(26^2) / (1 + r + r^2)^2]r + [(26^2) / (1 + r + r^2)^2](r^2) + [(26^2) / (1 + r + r^2)^2](r^3) = 156
[676r / (1 + r + r^2)^2] + [676r^2 / (1 + r + r^2)^2] + [676r^3 / (1 + r + r^2)^2] = 156
Step 5: Multiply both sides of the equation by (1 + r + r^2)^2 to eliminate the denominator:
676r + 676r^2 + 676r^3 = 156(1 + r + r^2)^2
Step 6: Simplify the equation further:
676r + 676r^2 + 676r^3 = 156(1 + 2r + r^2)
Step 7: Expand the right side of the equation:
676r + 676r^2 + 676r^3 = 156 + 312r + 156r^2
Step 8: Rearrange the equation and combine like terms:
676r^3 + 520r^2 + 364r - 156 - 312r = 0
676r^3 + 208r^2 + 52r - 156 = 0 ---> Equation (2)
Now, we have obtained a cubic equation (Equation 2) that we need to solve to find the values of r.
Step 9: Solve the cubic equation for r.
Unfortunately, solving a cubic equation analytically can be complicated and may require advanced methods. However, there are numerical methods like graphing or using software such as MATLAB or Wolfram Alpha that can be used to find approximate solutions. These methods can provide numerical values for r.
Once we have the values of r, we can substitute them back into Equation (1) to find the corresponding values of p using the formula:
p = 26 / (1 + r + r^2)
Therefore, by solving the cubic equation (Equation 2) for r and substituting the obtained values of r into Equation (1), we can determine all the real values of p and r that satisfy the given system of equations.