Question

已知4x+5y-7z=0 x+2y-z=0,且xyz≠0，试求x²+2y²-6z²/2x²-9y²-4z²的值。

已知4x+5y-7z=0 x+2y-z=0,且xyz≠0，试求x²+2y²-6z²/2x²-9y²-4z²的值。

Answer 1

4x+5y-7z=0 .................1
x+2y-z=0....................2
2式*4-1式得
3y+3z=0
y+z=0
y=-z......................3
将3式代入2式得
x+2*(-z)-z=0
x-3z=0
x=3z

(x²+2y²-6z²)/(2x²-9y²-4z²)
=[(3z)²+2(-z)²-6z²]/[2(3z)²-9(-z)²-4z²]
=[9z²+2z²-6z²]/[18z²-9z²-4z²]
=(5z²)/(5z²)
=1

Answer 2

4x+5y-7z=0 ........................... ①
x+2y-z=0................................ ②
②×4-①得3y+3z=0, y=-z...........③
③代入②，x=3z....................④
把③和④代入x²+2y²-6z²/2x²-9y²-4z²
=（3z）²+2×(-z)²-6z²/2×（3z）²-9×(-z)²-4z²
=5z²/5z²
=1

Answer 3

4x+5y-7z=0
4(x+2y-z)=0
相减得y+z=0
得y=-z
代入x+2y-z=0得
x=-3y=3z代入得
x²+2y²-6z²/2x²-9y²-4z²=1