Evaluate x^2−2y^2+z^2.
Solution
We are given:
(1) 9x−10y+z=8
(2) x+8y−9z=10
To eliminate z, multiply (1) by 9 and add to (2) to get:
82x−82y=82
Which implies:
x=y+1
Substituting for x in (1), we get:
9(y+1)−10y+z=8 or z−y=−1 so y−z=1
putting this in (2), we find no contradiction.
Thus, we have:
(3) x−y=1=> x = y+ 1
(4) y−z=1=> z = y- 1
x^2−2y^2+z^2= (y+1) ^2 + (y-1)^2 – 2y ^2 = 2 after expanding and simplifying
(1) 9x−10y+z=8
(2) x+8y−9z=10
To eliminate z, multiply (1) by 9 and add to (2) to get:
82x−82y=82
Which implies:
x=y+1
Substituting for x in (1), we get:
9(y+1)−10y+z=8 or z−y=−1 so y−z=1
putting this in (2), we find no contradiction.
Thus, we have:
(3) x−y=1=> x = y+ 1
(4) y−z=1=> z = y- 1
x^2−2y^2+z^2= (y+1) ^2 + (y-1)^2 – 2y ^2 = 2 after expanding and simplifying
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