F.3 Maths (linear equations)

1.
3^(x+y)=81......(1)
81^(x-y)=3.......(2)


From (2),
81^(x-y)=3
(3^4)^(x-y)=3
3^(4(x-y))=3
3^(4x-4y)=3......(3)

(1)*(3):
3^(x+y)*3^(4x-4y)=81*3
3^(x+y+4x-4y)=3^5
3^(5x-3y)=3^5
5x-3y=5...........(4)

From (1),
3^(x+y)=81
3^(x+y)=3^4
x+y=4
x=4-y..............(5)

Sub. (5) into (4),
5(4-y)-3y=5
20-5y-3y=5
-8y=-15
y=15/8

Sub. y=15/8 into (5)
x=4-15/8
x=17/8

∴x=17/8, y=15/8


2a.
7m+4n=18.......(1)
7m-6n=8..........(2)

(1)-(2):
10n=10
n=1

Sub. n=1 into (2),
7m+4(1)=18
7m=14
m=7

∴n=1, m=7



2010-10-03 19:14:28 補充：
2b.
7x+4y=18xy.........(1)
7x-6y=8xy ...........(2)

From (1),
7x+4y=18xy
(7x+4y)/xy=18xy/xy
(7x/xy)+(4y/xy)=18
7/y+4/x=18..........(3)

From (2),
7x-6y=8xy
(7x-6y)/xy=8xy/xy
(7x/xy)-(6y/xy)=8
7/y-6/x=8............(4)

So we have:
7/y+4/x=18..........(3)
7/y-6/x=8.............(4)

2010-10-03 19:14:35 補充：
Let m=1/y and n=1/x,
7m+4n=18.......(3)
7m-6n=8..........(4)

From 2a,
n=1
1/x=1
x=1

m=2
1/y=2
y=1/2

2010-10-03 19:15:47 補充：
更正:
2a .
Sub. n=1 into (2),
7m+4(1)=18
7m=14
m=2

∴n=1, m=2

1. 3^(x+y) = 81 ---(1)
81^(x-y) = 3 ---(2)
From (1), 3^(x+y) = 3^4
x+y = 4 ---(3)
From (2), 3^4(x-y) = 3
4(x-y) = 3 ---(4)
(3)*4, 4x+4y = 16 ---(5)
(5)-(3), 8y = 13
y = 13/8
Put y = 13/8 into (3), x = 19/8

2a. 7m+4n = 18 ---(1)
7m-6n = 8 ---(2)
(1)-(2), 10n = 10
n = 1
Put n = 1 into (1),
m = (18-4)/7
m = 2

b. Put x = m and y = n
So we have n = mn and m = 2mn
Hence, m = 1 and n = 1/2
and x = 1 and y = 1/2