SOLUTION: How do I solve this system? {{{ x^2 -xy +8=0 }}} {{{ x^2 - 8x +y = 0 }}}

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Question 1081674: How do I solve this system?
+x%5E2+-xy+%2B8=0+
+x%5E2+-+8x+%2By+=+0+
Found 2 solutions by josgarithmetic, solver91311:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
y=8x-x%5E2
-
x%5E2-x%288x-x%5E2%29%2B8=0
x%5E2-8x%5E2%2Bx%5E3%2B8=0
-7x%5E2%2Bx%5E3%2B8=0
x%5E3-7x%5E2%2B8=0

The one rational root for the cubic equation seems to be -1. Check on this further and if it works, find the resulting roots for the quadratic factor part. Continue to find the corresponding y values.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Solve both equations for in terms of

Equation 1:







Equation 2:



Now that you have two expressions in both equal to , set the two expressions equal to each other:







The rational roots theorem tells us that if a rational root exists, it must be one of the following values:

Use Synthetic Division: (that the value 1 fails is left as an exercise for the student) 
-1   |   1    -7    0    8
              -1    8   -8
     ---------------------
         1    -8    8    0

Hence, and are factors of the cubic.

Therefore one of the roots is -1 and the other two are the roots of the quadratic factor (calculation of these roots is left as an exercise for the student).

Using with




One of the three points of intersection is .

The other two are left for you to calculate.


John

My calculator said it, I believe it, that settles it