SOLUTION: Translate the axes to eliminate the first degree terms of {{{3x^2+xy+y^2-16x-10y+30=0}}} Thank you.

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Question 1141608: Translate the axes to eliminate the first degree terms of
3x%5E2%2Bxy%2By%5E2-16x-10y%2B30=0 Thank you.

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2%2Bxy%2By%5E2-16x-10y%2B30=0

We replace x by (x+h) and y by (y+k)



There is no need to multiply everything out yet, so we just list 
the first degree terms:

6hx%2Bkx%2Bhy%2B2ky-16x-10y

Group the x terms and the y-terms and factor out x and y:

%286h%2Bk-16%29x%2B%28k%2B2k-10%29y

The first degree terms will vanish (become 0) if the coefficients of x and y
are both 0.  So we set them equal 0:

system%286h%2Bk-16=0%2Ch%2B2k-10=0%29 

We solve that system by elimination or substitution
and get h=2, k=4



3%28x%5E2%2B4x%2B4%29%2B%28xy%2B4x%2B2y%2B8%29%2B%28y%5E2%2B8y%2B16%29-16x-32-10y-40%2B30=0

3x%5E2%2B12x%2B12%2Bxy%2B4x%2B2y%2B8%2By%5E2%2B8y%2B16-16x-32-10y-40%2B30=0

That simplifies to

3x%5E2+%2B+xy+%2B+y%5E2+-+6+=+0

So we shift the graph 2 units left and 4 units down and that 
will eliminate the first degree terms, leaving only terms of
degrees 0 and 2.

Edwin