📚 Solving Linear Equations and Simplifying Expressions | 解一元一次方程与化简表达式
This revision guide covers the core algebra techniques needed for Cambridge KS3 Mathematics, especially simplifying expressions, expanding brackets and solving linear equations. It is designed to help you build confidence with the type of questions found in page 95, Exercise 2, where unknowns appear on one or both sides of an equation.
本复习指南涵盖剑桥 KS3 数学所需的核心代数技能，特别是化简表达式、展开括号以及解一元一次方程。它旨在帮助你自信地应对第 95 页练习 2 中出现的题型，包括未知数出现在方程一边或两边的情况。
1. Algebraic Terms and Expressions | 代数项与表达式
An algebraic expression is a combination of numbers, letters and operation signs. For example, 3x + 5 and 2a - 7b are expressions. A term is a single number, a single letter, or numbers and letters multiplied together, such as 4x, -2y or 7. The number in front of a letter is called the coefficient. In 4x, the coefficient is 4.
代数表达式是由数字、字母和运算符号组成的式子。例如 3x + 5 和 2a - 7b 都是表达式。项是一个数、一个字母，或数字与字母相乘得到的整体，如 4x、-2y 或 7。字母前面的数叫作系数。在 4x 中，系数是 4。
To work confidently with equations, you need to recognise like terms. Like terms contain exactly the same letters and powers. For example, 3x and -5x are like terms, but 3x and 3x² are not like terms because the powers of x are different.
要自信地处理方程，你需要识别同类项。同类项包含完全相同的字母和幂。例如 3x 和 -5x 是同类项，但 3x 和 3x² 不是同类项，因为 x 的幂不同。
2. Collecting Like Terms | 合并同类项
ariable part unchanged. For example, 3x + 5x - 2x simplifies to 6x, because 3 + 5 - 2 = 6.
化简表达式通常意味着合并同类项。你只需将系数相加或相减，变量部分保持不变。例如，3x + 5x - 2x 化简为 6x，因为 3 + 5 - 2 = 6。
lecting the a terms and the b terms: 2a - a = a and 3b + 4b = 7b, so the answer is a + 7b.
当一个表达式含有不同类型的项时，应分别合并。化简 2a + 3b - a + 4b 时，先合并 a 项和 b 项：2a - a = a，3b + 4b = 7b，所以结果是 a + 7b。
- Example: Simplify 5m + 2n - 3m + 6n - 4.
- 示例：化简 5m + 2n - 3m + 6n - 4。
- 2m, 2n + 6n = 8n, and the constant -4 stays alone, so the simplified form is 2m + 8n - 4.
- 5m - 3m = 2m，2n + 6n = 8n，常数 -4 保持不变，因此化简结果为 2m + 8n - 4。
3. Expanding Brackets | 展开括号
term outside. For example, 3(x + 4) expands to 3x + 12, because 3 × x = 3x and 3 × 4 = 12.
展开括号是指将括号外的数或项与括号内的每一项相乘。例如，3(x + 4) 展开为 3x + 12，因为 3 × x = 3x，3 × 4 = 12。
0, because -2 × x = -2x and -2 × -5 = +10. This is a common place where sign errors occur.
当括号外的数是负数时要特别小心。展开 -2(x - 5) 得到 -2x + 10，因为 -2 × x = -2x，而 -2 × -5 = +10。这是符号错误常见的地方。
ch term: 4 × 2x = 8x, 4 × 3y = 12y, and 4 × -1 = -4, so the expanded form is 8x + 12y - 4.
对于含有两个以上项的括号，如 4(2x + 3y - 1)，将每一项分别相乘：4 × 2x = 8x，4 × 3y = 12y，4 × -1 = -4，所以展开结果为 8x + 12y - 4。
4. Solving One-Step Equations | 解一步方程
A linear equation states that two expressions are equal, and it contains an unknown, usually written as x. To solve an equation, you need to find the value of the unknown that makes the equation true. Always keep the equation balanced by doing the same operation to both sides.
一元一次方程表示两个表达式相等，并含有一个未知数，通常写作 x。要解方程，你需要找出使方程成立的未知数的值。始终对方程两边同时进行相同的运算，以保持方程平衡。
+ 3 - 3 = 7 - 3, so x = 4. If 5x = 35, divide both sides by 5: 5x ÷ 5 = 35 ÷ 5, so x = 7.
对于一步方程，使用逆运算。如果 x + 3 = 7，两边同时减去 3：x + 3 - 3 = 7 - 3，所以 x = 4。如果 5x = 35，两边同时除以 5：5x ÷ 5 = 35 ÷ 5，所以 x = 7。
x + 3 = 7 ⟶ x = 7 - 3 ⟶ x = 4
5x = 35 ⟶ x = 35 ÷ 5 ⟶ x = 7
5. Solving Two-Step Equations | 解两步方程
rst multiplied by 2 and then 3 is added. To solve it, reverse the order of operations: subtract 3 first, then divide by 2.
两步方程涉及对未知数进行两次运算。例如，在 2x + 3 = 11 中，未知数 x 先乘以 2，然后加上 3。解方程时要逆着运算顺序进行：先减去 3，再除以 2。
2x + 3 = 11 ⟶ 2x = 11 - 3 ⟶ 2x = 8 ⟶ x = 8 ÷ 2 ⟶ x = 4
on by 3 and subtraction of 5. Add 5 to both sides first: 3x = 21, then divide by 3: x = 7.
在方程 3x - 5 = 16 中，运算是乘以 3 和减去 5。首先两边同时加上 5：3x = 21，然后除以 3：x = 7。
3x - 5 = 16 ⟶ 3x = 16 + 5 ⟶ 3x = 21 ⟶ x = 7
back into the original equation. If x = 7, then 3 × 7 - 5 = 21 - 5 = 16, which is correct.
始终将答案代回原方程进行检验。如果 x = 7，那么 3 × 7 - 5 = 21 - 5 = 16，结果正确。
6. Equations with Brackets | 含括号的方程
two-step equation: subtract 6 from both sides to get 2x = 8, and divide by 2 to get x = 4.
当方程含有括号时，首先要展开括号。例如，2(x + 3) = 14 应先展开为 2x + 6 = 14。然后按照两步方程来解：两边同时减去 6 得到 2x = 8，再除以 2 得到 x = 4。
2(x + 3) = 14 ⟶ 2x + 6 = 14 ⟶ 2x = 8 ⟶ x = 4
= 21, first expand to 6x - 3 = 21. Add 3 to both sides: 6x = 24. Then divide by 6: x = 4.
如果方程是 3(2x - 1) = 21，先展开为 6x - 3 = 21。两边同时加上 3：6x = 24。然后除以 6：x = 4。
3(2x - 1) = 21 ⟶ 6x - 3 = 21 ⟶ 6x = 24 ⟶ x = 4
ill expand it as 15 = 5x - 10. Then add 10 to both sides: 25 = 5x, and divide by 5: x = 5.
有时括号在右边，例如 15 = 5(x - 2)。你仍然可以将其展开为 15 = 5x - 10。然后两边同时加上 10：25 = 5x，再除以 5：x = 5。
7. Unknowns on Both Sides | 两边均含未知数
h sides: 2x + 4 = 10. Then subtract 4 from both sides: 2x = 6. Finally divide by 2: x = 3.
有些方程两边都含未知数项，例如 3x + 4 = x + 10。先把 x 项移到一边，数字项移到另一边。两边同时减去 x：2x + 4 = 10。然后两边同时减去 4：2x = 6。最后除以 2：x = 3。
3x + 4 = x + 10 ⟶ 2x + 4 = 10 ⟶ 2x = 6 ⟶ x = 3
ct 2x from both sides to get 3x - 6 = 9. Add 6 to both sides: 3x = 15. Divide by 3: x = 5.
对于 5x - 6 = 2x + 9，两边同时减去 2x 得到 3x - 6 = 9。两边同时加上 6：3x = 15。除以 3：x = 5。
5x - 6 = 2x + 9 ⟶ 3x - 6 = 9 ⟶ 3x = 15 ⟶ x = 5
. For example, 8 = 2x - 4 becomes 2x - 4 = 8. Then add 4 to both sides: 2x = 12, so x = 6.
如果未知数出现在右边，而你希望它在左边，可以在第一步交换两边。例如，8 = 2x - 4 可以写成 2x - 4 = 8。然后两边同时加上 4：2x = 12，所以 x = 6。
8. Equations with Fractions | 含分数的方程
lve this, subtract 2 from both sides first: x/3 = 3. Then multiply both sides by 3: x = 9.
方程中可能含有分数，例如 x/3 + 2 = 5。解这个方程时，先两边同时减去 2：x/3 = 3。然后两边同时乘以 3：x = 9。
x/3 + 2 = 5 ⟶ x/3 = 3 ⟶ x = 3 × 3 ⟶ x = 9
(2x)/5 = 4, multiply both sides by 5 first: 2x = 20. Then divide both sides by 2: x = 10.
对于 (2x)/5 = 4，先两边同时乘以 5：2x = 20。然后两边同时除以 2：x = 10。
(2x)/5 = 4 ⟶ 2x = 4 × 5 ⟶ 2x = 20 ⟶ x = 10
as x/4 - 3 = 1, add 3 to both sides first: x/4 = 4. Then multiply both sides by 4: x = 16.
如果分数是被减去的，例如 x/4 - 3 = 1，先两边同时加上 3：x/4 = 4。然后两边同时乘以 4：x = 16。
x/4 - 3 = 1 ⟶ x/4 = 4 ⟶ x = 16
9. Checking Solutions and Inverse Operations | 检验解与逆运算
2(x + 3) = 14 and you found x = 4, then 2(4 + 3) = 2 × 7 = 14, so the solution is correct.
解完方程后，一定要将答案代回原方程，检验两边是否相等。如果原方程是 2(x + 3) = 14，而你得到 x = 4，那么 2(4 + 3) = 2 × 7 = 14，所以解是正确的。
If the sides are not equal, go back and check each step. Common errors include adding when you should subtract, forgetting to apply an operation to both sides, or making a sign mistake when expanding brackets.
如果两边不相等，请返回去检查每一步。常见错误包括该减的时候加了、忘记对两边同时进行同一操作，或展开括号时出现符号错误。
Remember that addition and subtraction are inverse operations, and multiplication and division are inverse operations. Use the inverse operation to undo what has been done to the unknown.
记住，加法和减法是互逆运算，乘法和除法也是互逆运算。使用逆运算来撤销对未知数所做的运算。
| Operation | Inverse Operation |
|---|---|
| + 3 | - 3 |
| - 5 | + 5 |
| × 2 | ÷ 2 |
| ÷ 4 | × 4 |
10. Solving Word Problems with Linear Equations | 用一元一次方程解应用题
Many KS3 questions ask you to form an equation from a written problem. Start by choosing a letter for the unknown quantity. Then translate the words into algebra, write an equation, solve it and check your answer makes sense.
许多 KS3 题目要求你根据文字问题列出方程。首先选择一个字母表示未知量。然后把文字翻译成代数式，写出方程，解方程，并检查答案是否合理。
For example: "I think of a number, multiply it by 4 and add 7. The result is 27." Let the number be n. Then 4n + 7 = 27. Subtract 7: 4n = 20. Divide by 4: n = 5.
例如：“我想一个数，把它乘以 4，再加上 7。结果是 27。”设这个数为 n。那么 4n + 7 = 27。减去 7：4n = 20。除以 4：n = 5。
4n + 7 = 27 ⟶ 4n = 20 ⟶ n = 5
Another example: "Three more than twice a number is 15." than that is 2x + 3. The equation is 2x + 3 = 15. Subtract 3: 2x = 12. Divide by 2: x = 6.
另一个例子：“一个数的两倍再加 3 等于 15。”设这个数为 x。它的两倍是 2x，再加 3 就是 2x + 3。方程为 2x + 3 = 15。减去 3：2x = 12。除以 2：x = 6。
11. Common Mistakes to Avoid | 常见错误与避免方法
t 5 from both sides, not just the left side. The correct step is x = 12 - 5, giving x = 7.
一个常见错误是忘记必须对两边同时进行同一运算。例如，在 x + 5 = 12 中，你必须两边同时减去 5，而不是只从左边减。正确的步骤是 x = 12 - 5，得到 x = 7。
Another mistake is mishandling negative signs when expanding brackets. Expanding -3(x - 2) gives -3x + 6, not -3x - 6. Pay attention to the sign of every term inside the bracket.
另一个错误是展开括号时错误处理负号。展开 -3(x - 2) 得到 -3x + 6，而不是 -3x - 6。要注意括号内每一项的符号。
Some students also combine unlike terms by mistake. For example, 2x + 3 cannot be simplified to 5x, because 2x and 3 are not like terms. Only terms with the same variable part can be combined.
有些学生还会错误地合并不是同类项的项。例如，2x + 3 不能化简为 5x，因为 2x 和 3 不是同类项。只有变量部分相同的项才能合并。
12. Practice Questions and Summary | 练习题与总结
3a - 2b + 5a + 6b - 7; Solve 5(x - 3) = 20; Solve 6x - 4 = 2x + 8; and Solve x/5 + 2 = 7.
尝试以下题目来检验你的理解：解 4x + 9 = 29；化简 3a - 2b + 5a + 6b - 7；解 5(x - 3) = 20；解 6x - 4 = 2x + 8；解 x/5 + 2 = 7。
8, add 4: 4x = 12, so x = 3. For x/5 + 2 = 7, subtract 2: x/5 = 5, multiply by 5: x = 25.
对于 4x + 9 = 29，减去 9：4x = 20，然后除以 4：x = 5。对于 3a - 2b + 5a + 6b - 7，合并同类项：8a + 4b - 7。对于 5(x - 3) = 20，展开：5x - 15 = 20，加上 15：5x = 35，所以 x = 7。对于 6x - 4 = 2x + 8，减去 2x：4x - 4 = 8，加上 4：4x = 12，所以 x = 3。对于 x/5 + 2 = 7，减去 2：x/5 = 5，乘以 5：x = 25。
To master linear equations, practise the flow: simplify both sides, collect like terms, undo operations in reverse order, and always check your solution. With consistent practice, these algebra skills become automatic and reliable for exams.
要掌握一元一次方程，请反复练习以下流程：化简两边、合并同类项、逆着运算顺序求解，并始终检验答案。通过持续练习，这些代数技能将变得熟练且可靠，帮助你在考试中稳定发挥。
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