📚 Solving Linear Equations with Unknowns on Both Sides | 求解含两边未知量的一次方程
Being able to solve equations where the variable appears on both sides of the equals sign is a key milestone in Cambridge KS3 mathematics. It tests your understanding of balance, inverse operations, and algebraic manipulation in one go.
能够求解变量出现在等号两边的一次方程，是剑桥 KS3 数学中的一个关键里程碑。它同时考查你对平衡关系、逆运算和代数变形的理解。
This revision guide focuses on the exact skill needed for a typical Cambridge KS3 exercise such as page 44, question 2. Work through each worked example carefully before trying the practice questions.
本复习指南重点讲解剑桥 KS3 典型练习（如第44页第2题）所需的技能。请先认真阅读每个例题，再尝试后面的练习题。
1. Why This Topic Matters | 为什么这个主题重要
In KS3 algebra, solving equations with unknowns on both sides is the bridge between simple one-step equations and more complex problems involving brackets, fractions, and inequalities. Cambridge assessments often include this type because it checks whether you can think flexibly rather than just follow a fixed rule.
在 KS3 代数中，求解两边都有未知量的方程是连接一步方程与更复杂问题（含括号、分数和不等式）的桥梁。剑桥评估通常包含这种类型，因为它考查你是否能灵活思考，而不只是遵循固定规则。
You will also meet the same skill in real-life problems such as comparing mobile phone tariffs, calculating break-even points, or matching travel distances. If you can balance an equation confidently, you can model and solve many practical situations.
你还会在现实生活中遇到相同的技能，例如比较手机话费套餐、计算盈亏平衡点或匹配行驶距离。如果你能熟练平衡方程，就可以建立并解决许多实际问题。
2. Key Vocabulary and Symbols | 关键术语与符号
Before solving, make sure you know the words 'equation', 'expression', 'variable', 'term', 'coefficient', 'constant', and 'solution'. An equation says two expressions are equal; a solution is the value that makes the equality true.
在求解之前，请确保你认识这些词：equation（方程）、expression（表达式）、variable（变量）、term（项）、coefficient（系数）、constant（常数）和 solution（解）。方程表示两个表达式相等；解就是使等号成立的值。
The balancing method means you must do the same operation to both sides. Useful inverse pairs are addition/subtraction and multiplication/division. We will also use the symbol '⇒' to mean 'implies' when showing the next step.
平衡法指你必须对等号两边同时进行相同的运算。常用的逆运算对包括加法/减法和乘法/除法。我们还会使用符号 '⇒' 表示“推出”或“下一步”。
3x + 2 = x + 10
In this equation, 3x and x are variable terms, 2 and 10 are constants, and the solution is the value of x that balances both sides.
在这个方程中，3x 和 x 是含变量项，2 和 10 是常数项，解就是使两边平衡的 x 值。
3. Balancing Method Review | 平衡法回顾
The balancing method is based on a simple idea: if you add, subtract, multiply, or divide one side of a true equation, you must do exactly the same to the other side. Think of a balanced seesaw with equal weights on both sides.
平衡法基于一个简单的思想：如果你对方程的一边进行加、减、乘、除，就必须对另一边做完全相同的运算。想象一个两边重量相等的跷跷板。
to get x = 5. These one-step moves prepare you for equations with variables on both sides.
例如，解方程 x + 5 = 11 时，两边同时减去 5，得到 x = 6。解方程 4x = 20 时，两边同时除以 4，得到 x = 5。这些一步运算可以为你解两边都有变量的方程做好准备。
When the unknown appears on both sides, the first goal is always to collect all variable terms on one side and all constant terms on the other. After that, the equation becomes a simple one-step or two-step equation.
当未知量出现在两边时，第一个目标始终是把所有含变量的项移到一边，把所有常数项移到另一边。之后，方程就变成简单的一步或两步方程。
4. Identifying Unknowns on Both Sides | 识别两边都有未知数的方程
case, the same variable appears on the left-hand side (LHS) and the right-hand side (RHS).
注意像 3x + 2 = x + 10、5x − 4 = 2x + 8 或 7 − x = 2x + 1 这样的方程。在每种情况下，同一个变量都出现在左边（LHS）和右边（RHS）。
ove everything at once. Choose the smaller variable term first. For example, in 3x + 2 = x
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