📚 Solving Linear Equations | 解一元一次方程
Linear equations are at the heart of algebra in Key Stage 3. In Cambridge Checkpoint Mathematics, you will learn how to find unknown values by keeping both sides of an equation balanced. Mastering this skill builds a solid foundation for all future work in algebra, graphs and problem solving.
线性方程是 KS3 代数的核心。在 Cambridge Checkpoint 数学中，你将学习如何通过保持等式两边平衡来求出未知数。掌握这一技能会为今后的代数、图像和问题解决打下坚实基础。
1. What is a Linear Equation? | 什么是一元一次方程？
er of one. There are no ² or ³ terms. Examples include x + 3 = 7, 2x = 10 and 4x − 1 = 11.
一元一次方程是一个代数等式，其中未知数（通常为 x）的最高次数为 1，没有 ² 或 ³ 这样的项。例子包括 x + 3 = 7，2x = 10 和 4x − 1 = 11。
true. For x + 3 = 7, subtracting 3 shows that x = 4. The equation is like a balanced scale – whatever you do to one side, you must do to the other.
方程的解就是使等式成立的 x 值。对于 x + 3 = 7，两边减 3 得到 x = 4。方程就像一架平衡的天平——你对一边做什么，对另一边也必须做同样的事。
2. The Balancing Method | 平衡法
The balancing method is the key to solving all linear equations. Imagine a pair of scales in equilibrium. If you add, subtract, multiply or divide on one side, you must perform exactly the same operation on the other side to keep the scales balanced.
平衡法是解所有一元一次方程的关键。想象一架处于平衡状态的天平。如果你在一侧加、减、乘或除，就必须在另一侧进行完全相同的运算，以保持天平平衡。
This idea can be written as: whatever you do to the left-hand side (LHS), do to the right-hand side (RHS). This preserves the equality and steadily isolates the unknown.
这个想法可以写成：你对左边（LHS）做什么，就对右边（RHS）做什么。这样做能保持相等，并逐步把未知数分离出来。
3. Solving One-Step Equations | 解一步方程
both sides: x = 7. For subtraction equations like x − 3 = 9, add 3 to both sides: x = 12.
一步方程只需一次运算就能求出 x。对于 x + 5 = 12 这样的加法方程，两边同减 5：x = 7。对于 x − 3 = 9 这样的减法方程，两边同加 3：x = 12。
by multiplying both sides by 4: x = 8. Always check your answer in the original equation.
像 3x = 18 这样的乘法方程，两边同除以 3：x = 6。像 x ÷ 4 = 2 这样的除法方程，两边同乘 4：x = 8。一定要把答案代回原方程检验。
4. Solving Two-Step Equations | 解两步方程
2x = 8, then divide both sides by 2, giving x = 4. Always reverse the order of operations – deal with addition or subtraction before multiplication or division.
两步方程包含两种运算。对于 2x + 3 = 11，先两边同减 3 得到 2x = 8，然后两边同除以 2，得出 x = 4。一定要逆序处理——先处理加减，再处理乘除。
then subtract 1: 4 = 2x, so x = 2. Planning your steps carefully can prevent sign errors.
如果方程是 5 − 2x = 1，你可以两边同时加上 2x 来避免负系数：5 = 1 + 2x，然后减 1：4 = 2x，所以 x = 2。仔细规划步骤可以防止符号错误。
5. Equations with Brackets | 含括号的方程
de by 3: x = 3. Expanding simplifies the equation to a form you already know how to solve.
当方程含有括号时，要先将括号展开。对于 3(x + 2) = 15，乘开得到 3x + 6 = 15。然后减 6：3x = 9，再除以 3：x = 3。展开后方程就化为了你会解的形式。
ive law: a(b + c) = ab + ac. Work carefully with negative signs, e.g. −2(x − 3) = −2x + 6.
有时一个数字乘在括号内的和或差上。记住分配律：a(b + c) = ab + ac。处理负号时要仔细，例如 −2(x − 3) = −2x + 6。
6. Equations with Unknowns on Both Sides | 未知数在两侧的方程
2x = 6, so x = 3. The goal is to gather all x terms on one side and numbers on the other.
当方程两边都含有 x 时，要合并同类项。对于 5x + 2 = 3x + 8，两边同减 3x 得到 2x + 2 = 8。然后减 2：2x = 6，所以 x = 3。目的是把所有 x 项集中到一边，数字移到另一边。
x: 4x − 4 = 10, add 4: 4x = 14, x = 3.5. Always aim to make the coefficient of x positive.
如果一边有多个含 x 的项，先将它们化简。对于 2x + 3x − 4 = x + 10，合并成 5x − 4 = x + 10，然后减 x：4x − 4 = 10，加 4：4x = 14，x = 3.5。尽量让 x 的系数保持正数。
7. Dealing with Fractions | 处理分数
6 = 10, so x = 4. This technique turns fractional equations into simple integer equations.
可以通过各项乘以最小公分母来消去分数。对于 ½x + 3 = 5，每项乘以 2：x + 6 = 10，所以 x = 4。这种方法可以把分数方程化成简单的整数方程。
. When fractions have different denominators, find the LCM of all denominators and multiply every term by it.
若方程为 (x/3) − 2 = 1，两边同乘 3：x − 6 = 3，所以 x = 9。当分母不同时，先求出所有分母的最小公倍数，然后将每一项乘以它。
8. Forming Equations from Word Problems | 从文字问题建立方程
Real-life problems often require you to write an equation before solving. Read the question carefully and let x represent the unknown quantity. Translate words like 'sum' to addition, 'difference' to subtraction, 'product' to multiplication and 'quotient' to division.
现实问题往往需要先列出方程再求解。仔细读题，用 x 表示未知量。把“和”翻译为加法，“差”为减法，“积”为乘法，“商”为除法。
For example, 'Three more than twice a number is 11' becomes 2x + 3 = 11. 'The perimeter of a square is 20 cm' becomes 4x = 20 if side length is x. Always define x clearly before writing the equation.
例如，“一个数的两倍加 3 等于 11”可写成 2x + 3 = 11。“正方形周长为 20 厘米”，设边长为 x，则 4x = 20。在列方程前一定要明确 x 表示什么。
9. Checking Your Solution | 检验你的解
lution is x = 4. LHS: 4(4) − 5 = 16 − 5 = 11, which equals RHS. This confirms correctness.
解完后，一定要把求得的 x 值代回原方程验证是否正确。对于 4x − 5 = 11，解为 x = 4。左边：4(4) − 5 = 16 − 5 = 11，等于右边，这就证实了正确性。
Checking helps you catch arithmetic errors or sign mistakes. In exams, it can earn you marks even if your working has a slip. Make substitution a habit.
检验可以帮助你发现计算错误或符号错误。在考试中，即使解题过程有小失误，检验也可能为你赢得分数。要养成代入验证的习惯。
10. Common Mistakes to Avoid | 常见错误及避免方法
x. For ½x + 3 = 7, the correct step is 2 × (½x) + 2 × 3 = 2 × 7, giving x + 6 = 14, x = 8.
一个常见错误是忘记对每一项进行运算。如果两边同时乘 2，一定要乘以每一项，包括不含 x 的数字。对于 ½x + 3 = 7，正确步骤是 2 × (½x) + 2 × 3 = 2 × 7，得到 x + 6 = 14，x = 8。
Another mistake is mishandling negative signs. When moving a term to the other side, its sign flips. Also, expanding −(x − 4) yields −x + 4, not −x − 4. Slow down and be systematic.
另一个错误是负号处理不当。将一项移到另一边时，它的符号要改变。还有，将 −(x − 4) 展开得到 −x + 4，而不是 −x − 4。放慢速度，按步骤来。
11. Practice and Revision Tips | 练习与复习技巧
Start with simple one-step equations and gradually work up to more complex ones with brackets and unknowns on both sides. Use flashcards or online quizzes to speed up your mental arithmetic. Write each step clearly, showing how the equation changes.
先从简单的一步方程开始，逐步过渡到含有括号和两边都有未知数的复杂方程。使用记忆卡片或在线测验提高心算速度。每一步写清楚，展示方程的变化过程。
Redo questions you got wrong after a few days. Explain the steps to a friend – teaching someone else deepens your understanding. Aim to solve at least ten mixed equations a day to build fluency.
几天后重做做错的题目。向朋友讲解解题步骤——教别人能加深你的理解。每天尽量解至少 10 个混合方程，培养熟练度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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