SOLUTION: ax + by = 5 ax² + by² = 10 ax³ + by³ = 50 ax⁴ + by⁴ = 130 Find, 13(x + y - xy) - 120(a + b) = ?

There may be some trick that gives an easier way than to solve for all the
letters, but here it is that way: 


Solve the first two equations for a, by Cramer's rule:


Solve the 2nd and 3rd equations for a, by Cramer's rule:


Set the first and second values for a equal:

Multiply both sides by



equation 1:    
-----------------------------
Solve the 3rd and 4th equations for a, by Cramer's rule:


Set the second and third values for a equal:

Multiply both sides by



equation 2:    
Set the values of x from equations 1 and 2 equal












x and y cannot be equal because the first two given equations
would be inconsistent. For if they were equal, we'd have

Nultiplying thru by x gives
 or that would give x=10, not what we got for y.
Therefore by symmetry of x and y, one of them has the + sign and the
other has the minus sign. So we choose them so that


and


Since

, so




, so


Rationalizing the denominator we get

By symmetry, or by substituting,
 

So



 



















Edwin