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# help!!

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In this multi-part problem, we will consider this system of simultaneous equations:

3x+5y-6z=2, (i)

5xy-10yz-6xz=-41,  (ii)

xyz=6.  (iii)

Let a=3x, b=5y, and c=-6z.

i figured out a so i won't post it

(b) Given that (x,y,z) is a solution to the original system of equations, determine all distinct possible values of x+y.

i'm only finding a few values of x+y, how do i find all of them?

Jun 6, 2020

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Replace x, y, and z in equations by a, b, and c:

a + b + c = 2

a*b/3 + b*c/3 + a*c/3 = -41.  So.  a*b + b*c + a*c = -123

-a*b*c/(3*5*6) = 6. So. a*b*c = -540

(t - a)(t - b)(t - c) = 0. ⇒ t^3 - (a + b + c)t^2 + (a*b + b*c + a*c)t - a*b*c = 0

Hence:  t^3 - 2t^2 - 123t + 540 = 0

Solving this gives: a = -12, b = 5, c = 9 or variants of these.

So:  x = a/3 → -4,   y = b/5 → 1,  z = -c/6 → -3/2

x + y = -3 This is just one possibility.  The others can be found by interchanging the values allocated to a, b and c.

Hope this helps, whymenotsmart:)

Jun 6, 2020