Question

已知

3

x+y

=

4

y+z

=

5

z+x

，則

x2+y2+z2

xy+yz+zx

=

 

．

Answer 1

分析：首先將已知

3

x+y

=

4

y+z

=

5

z+x

轉化為

x+y

3

=

y+z

4

=

x+z

5

，并令

x+y

3

=

y+z

4

=

x+z

5

=a，通過解方程組用a分別表示x、y、z的值．再代入原式計算．

解答：解：∵

3

x+y

=

4

y+z

=

5

z+x

，
∴

3 x+y = 4 y+z 3 x+y = 5 z+x 4 y+z = 5 z+x

?

y+z 4 = x+y 3 x+z 5 = x+y 3 x+z 5 = y+z 4

?

x+y

3

=

y+z

4

=

x+z

5

∴令

x+y

3

=

y+z

4

=

x+z

5

=a
則有   

x+y=3a          ① y+z=4a          ② x+z=5a          ③

  
由①+②+③得   x+y+z=6a     ④
由④-①得   z=3a，
同理解得x=2a，y=a
∴

x2+y2+z2

xy+yz+zx

=

(22+12+32)a2

(2×1+1×3+2×3)a2

=

14

11

故答案為

14

11

．

點評：本題考查分式的化簡求值．首先將已知

3

x+y

=

4

y+z

=

5

z+x

轉化為

x+y

3

=

y+z

4

=

x+z

5

，對于不確定的上述表達式，可令

x+y

3

=

y+z

4

=

x+z

5

=a，進而不難用a分別表示x、y、z的值，這是解決本題的關鍵．