Simultaneous Equations

1.
4c + 2d + 6 = 3c - 3d + 9 = 0

4c + 2d + 6 = 0 ...... [1]
3c - 3d + 9 = 0 ...... [2]

From [1] :
2c + d = -3 ...... [3]

From [2] :
c - d = -3 ...... [4]

[3] + [4] :
3c = -6
c = -2

Put c = -2 into [4] :
(-2) - d + 3 = 0
d = 1

Hence, c = -2, d = 1


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2.
2r - 2s + 13 = 5r = 3s + 9

2r - 2s + 13 = 5r ...... [1]
5r = 3s + 9 ...... [2]

From [1] :
3r + 2s = 13
9r + 6s = 39 ...... [3]

From [2] :
5r - 3s = 9
10r - 6s = 18 ...... [4]

[3] + [4] :
19r = 57
r = 3

Put r = 3 into [2] :
5(3) = 3s + 9
3s = 6
s = 2

Hence, r = 3, s = 2


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3.
5b + 2a - 5 = 5a - 2(1 - b) = 0

5b + 2a - 5 = 0 ...... [1]
5a - 2(1 - b) = 0 ...... [2]

From [1] :
2a + 5b = 5
4a + 10b = 10 ...... [3]

From [2] :
5a - 2 + 2b = 0
25a + 10b = 10 ...... [4]

[4] - [3] :
21a = 0
a = 0

Put a = 0 into [1] :
5b + 0 - 5 = 0
b = 1

Hence, a = 0, b = 1


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4.
r + s = 3r - s + 4 = 0

r + s = 0 ...... [1]
3r - s + 4 = 0 ...... [2]

[1] + [2] :
4r + 4 = 0
r = -1

Put r = -1 into [1] :
(-1) + s = 0
s = 1

Hence, r = -1, s = 1


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5.
3p - (q - 10) = p - q = 0

3p - (q - 10) = 0 ...... [1]
p - q = 0 ...... [2]

From [1] :
3p - q = -10 ...... [3]

[3] - [2] :
2p = -10
p = -5

Put p = -5 into [2] :
(-5) - q = 0
q = -5

Hence, p = -5, q = -5


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6.
x + 4y - 1 = 3(x - 2) - 2(y - 5) = 0

x + 4y - 1 = 0 ...... [1]
3(x - 2) - 2(y - 5) = 0 ...... [2]

From [1] :
x + 4y = 1 ...... [3]

From [2] :
3x - 6 - 2y + 10 = 0
6x - 4y = -8 ...... [4]

[3] + [4] :
7x = -7
x = -1

Put x = -1 into [3] :
(-1) + 4y = 1
y = 1/2

Hence, x = -1, y = 1/2


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7.
3a + 2b - 2 = 5a + 7b + 15 = 0

3a + 2b - 2 = 0 ...... [1]
5a + 7b + 15 = 0 ...... [2]

[1] * 7 :
21a + 14b = 14 ...... [3]

[2] * 2 :
10a + 14b = -30 ...... [4]

[3] - [4] :
11a = 44
a = 4

Put a = 4 into [1] :
3(4) + 2b - 2 = 0
2b = -10
b = -5

Hence, a = 4, b = -5

2012-12-09 00:13:48 補充：
For example : 2r - 2s + 13 = 5r = 3s + 9

There are 3 possible equations :
2r - 2s + 13 = 5r
5r = 3s + 9
2r - 2s + 13 = 3s + 9

You can choose any two. In my opinion, the two simpler equations should be chosen.

I am sorry that i want to ask you something ,What is the technique of setting the equations,for example
2r - 2s + 13 = 5r = 3s + 9
Why choosing the middle one,or just every equation can be randomly chosen by you,according to your needs of convenience?Is there any hard rule to rememberThx, urgent!