中五數學 會考題

解: (a) (2x - 7y) / (2x - 4y) = - (1 / 2)

2 (2x - 7y) = - (2x -4y)

4x - 14y = - 2x + 4y

4x + 2x = 4y + 14y

6x = 18y

x = 3y

Ans: x : y = 3 : 1

(b) [ By (a) ], x : y = 3 : 1

Let x = 3k, y = k ---------------------(1)

2x^2 - 12y = 3(x -1) -----------------(2)

Sub (1) into (2),

2(3k)^2 - 12(k) = 3(3k - 1)

18k^2 - 12k = 9k - 3

18k^2 - 21k + 3 = 0

i.e. 6k^2 - 7k + 1 = 0

( 6k-1 ) ( k-1 ) = 0

k = 1 / 6 or 1

-- When k = 1 / 6,

x= 3( 1/6) = 1/2

y = 1/6

-- When k = 1,

x = 3(1) = 3

y = 1

已知(2x - 7y) / (2x - 4y) = - (1 / 2)
(a)求x : y
(2x - 7y) / (2x - 4y) = - (1 / 2)
2x - 7y = - (2x - 4y) (1 / 2)
4x-14y = - (2x - 4y)........................
4x-14y = - 2x + 4y
6x = 18y
x=3y
x/y = 3

所以x : y = 3 : 1


(b)此外 , 若2x^2 - 12y = 3(x -1) , 求x及y
代入x=3y
2x^2 - 12y = 3(x -1)
2(3y)² - 12y = 3(3y -1)
2(9y²) -12y = 9y-3
18y² -12y =9y-3
18y² -21y+3=0
y=1 或 y=1/6

當y=1
2x² - 12(1) = 3(x -1)
2x² -12 =3x-3
2x² -3x -9=0
x=3 或 x=-3/2

當y=1/6
2x² - 12(1/6) = 3(x -1)
2x² -2 =3x-3
2x² -3x +1=0
x=1 或 x=1/2