解二元一次聯立方程式..(2題/10點)

1.
(4/x) + (1/y) = 1 ...... (1)
(2/x) - (3/y) = 4 ..... (2)

由(1):
4(1/x) + (1/y) = 1 ...... (3)

由(2):
2(1/x) - 3(1/y) = 4
4(1/x) - 6(1/y) = 8 ...... (4)

(3)-(4):
7(1/y) = -7
1/y = -1
y = -1

把 1/y = -1 代入(3):
2(1/x) + 3 = 4
x = 2

答案：x = 2, y = -1


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2.
(-4/a) + (1/b) = 1 ...... (1)
ab = 2 ...... (2)

由(1):
(-4b/ab) + (a/ab) = 1
(a - 4b)/ab = 1
ab = a - 4b ...... (3)

(2) = (3):
a - 4b = 2
a = 4b + 2 ...... (4)

把 (4) 代入 (2):
(4b + 2)b = 2
4b² + 2b = 2
2b² + b - 1 = 0
(2b - 1)(b + 1) = 0
b = 1/2 或 b = -1

把 b = 1/2 代入(2):
a(1/2) = 2
a = 4

把 b = -1 代入(2):
a(-1) = 2
a = -2

答案： a = 4, b = 1/2 或 a = -2, b =-1

1.
4/x + 1/y = 1 .... (1)
2/x - 3/y = 4 .... (2)

令 X = 1/x , Y = 1/y , 則 (1) & (2) 變成

4*X + Y = 1 .... (3)
2*X - 3*Y = 4 .... (4)

(3)*3 + (4) => 14*X = 7 => X = 1/2 => x = 2
(3) - (4)*2 => 7*Y = -7 => Y = -1 => y = -1

2.
-4/a + 1/b = 1 .... (1)
ab = 2 .... (2)

(1)*a*b => -4*b + a = ab = 2 => a = 4*b + 2 .... (3)

代入 (2) => (4*b + 2)*b = 2 => 2*b^2 + b - 1 = 0
=> b = 1/2 , -1

故

a = 4 , -2

答案不確定是不是對的

1.把(1)×3，得到X分之12+Y分之3=3
在用(1)式+(2)式
得X分之2+X分之12=7
X分之14=7
X=2
再把X帶入(1)，得Y=-1
2.把(2)式變成a=b分之2......(3)
把(3)式帶入(1)式
得b分之2分之-4+b分之1=1
-4×2分之b+b分之1=1
-2+b分之1=1
b分之1=3
3b=1，b=3分之1
再把b帶入(2)式
得3分之1×a=2，a=2×3=6