數學問題，二元一次聯立方程式

1.
x + y + 2 = 2x + y - 3 ... [1]
x + y + 2 = 2y + 4 ... [2]

由 [1]:
x = 5

把 x = 5 代入[2]:
5 + y + 2 = 2y + 4
y = 3


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2.
2x - y + 3 = x ... [1]
x = 4x + 3y - 3 ... [2]

由 [1]:
y = x + 3 ... [3]

把 [3] 代入 [2]:
x = 4x + 3(x + 3) - 3
x = 4x + 3x + 9 - 3
6x = -6
x = -1

把 x = -1代入 [3]:
y = -1 + 3
y = 2


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3.
4a + b - 5 = -2a ... [1]
a - 3b - 27 = -2a ... [2]

由[2] :
3a - 3b - 27 = 0
a - b - 9 = 0
b = a - 9 ... [3]

把 [3] 代入 [1]:
4a + (a - 9) - 5 = -2a
7a = 14
a = 2

把 a = 2 代入 [3]:
b = 2 - 9
b = -7


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4.
3x + 8 = 2x - 5y + 1 ... [1]
2x - 5y + 1 = x + 3y + 7 ... [2]

由 [1]:
x = -5y - 7 ... [3]

把 [3] 代入 [2]:
2(-5y - 7) - 5y + 1 = (-5y - 7) + 3y + 7
-10y - 14 - 5y + 1 = -5y - 7 + 3y + 7
13y = -13
y = -1

把 y = -1 代入 [3]:
x = -5(-1) - 7
x = -2


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5.
2x + y + 5 = 3x - y - 6 ... [1]
4x + 3y + 7 = 3x - y - 6 ... [2]

由 [1]:
-x + 2y = -11 ... [3]

由 [2]:
x + 4y = -13 ... [4]

[3] + [4]:
6y = -24
y = -4

把 y= -4 代入[4]:
x + 4(-4) = -13
x = 3


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6.
6x + 5y - 14 = -5x - 7y + 13 ... [1]
-5x - 7y + 13 = 4x - 3y + 20 ... [2]

由 [1]:
11x + 12y = 27 ... [3]

由 [2]:
9x + 4y = -7
27x + 12y = -21 ... [4]

[4] - [3]:
16x = -48
x = -3

把 x = -3 代入 [3]:
11(-3) + 12y = 27
12y = 60
y = 5


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7.
設分子為 n，分母為 d。

3n = 2d + 7 ... [1]
(n + 5)/(d + 5) = 4/5 ... [2]

由 [2]:
5(n + 5) = 4(d + 5)
5n + 25 = 4d + 20
5n - 4d = -5 ... [3]

由 [1]:
3n - 2d = 7
6n - 4d = 14 ... [4]

[4] - [3]:
n = 19

把 n = 16 代入 [1]:
3(19) = 2d + 7
2d = 50
d = 25

此分數 = 19/25


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8.
設拾元的個數為 n 個。
則壹元的個數 = 2n 個
而伍元的個數 = 2(2n) + 3 個 = 4n + 3 個

金額合計：
10n + 2n + 5(4n + 3) = 111
32n = 96
n = 3

硬幣的個數
= 3 + 2*3 + (4*3 + 3) 個
= 3 + 6 + 15 個
= 24 個


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9.
x + y = 400 ... [1]
x(99%) + y(7%) = 400(30%) ... [2]

由 [2]:
99x + 7y = 12000 ... [3]

[1] * 7 :
7x + 7y = 2800 ... [4]

[3] - [4]:
92x = 9200
x = 100

把 x = 100 代入 [1]:
(100) + y = 400
y = 300