📚 Solving Linear Equations: Cambridge KS3 Mathematics Revision | 解一元一次方程：剑桥 KS3 数学复习
Linear equations are one of the first major algebraic tools you meet in Key Stage 3. They let you change a word problem into a visual balance and find an unknown value step by step. Mastery of this topic is essential for Cambridge Checkpoint because later topics such as graphs, sequences, and problem solving all rely on fluent equation solving.
一元一次方程是你在 Key Stage 3 遇到的第一个重要代数工具。它们让你能把文字题转化为一架可视的天平，并逐步求出未知数的值。掌握这一主题对剑桥 Checkpoint 考试至关重要，因为后续的图像、数列和问题解决等内容都依赖于熟练地解方程。
1. What Is a Linear Equation? | 什么是一元一次方程？
or products of x with itself. Examples include x + 5 = 12, 3x − 4 = 11, and 2(x + 1) = 10.
一元一次方程是一种仅含有一个未知数（通常写作 x）的方程，且未知数只出现一次方。方程中没有 x²、x³、平方根或 x 与自身相乘的项。例如 x + 5 = 12、3x − 4 = 11 和 2(x + 1) = 10。
The word linear tells us that its graph is a straight line. For KS3, you will usually solve equations by rearranging rather than by drawing graphs, but knowing the connection helps you see why equations have one solution.
线性这个词告诉我们它的图像是一条直线。在 KS3 阶段，你通常通过变形来解方程，而不是画图，但了解这种联系有助于你明白为什么这类方程只有一个解。
2. The Balance Method | 平衡法
Think of an equation as a set of balance scales. The equals sign is the pivot. If you add, subtract, multiply, or divide both sides by the same non-zero amount, the scales stay balanced. This is the golden rule of equation solving.
把方程想象成一架天平。等号就是支点。如果你对方程两边同时加、减、乘或除以同一个非零的数，天平仍然保持平衡。这是解方程的金科玉律。
+ 7 = 15. To isolate x, subtract 7 from both sides: x + 7 − 7 = 15 − 7, which gives x = 8.
我们可以用平衡法来解 x + 7 = 15。为了单独留下 x，两边同时减去 7：x + 7 − 7 = 15 − 7，得到 x = 8。
x + 7 = 15
x = 15 − 7
x = 8
3. Inverse Operations | 逆运算
Every operation has an inverse. Addition and subtraction undo each other, and multiplication and division undo each other. When solving, use the inverse operation to remove a number from the side containing the variable.
每种运算都有逆运算。加法与减法互为逆运算，乘法与除法互为逆运算。解方程时，使用逆运算来去掉含有未知数那一边的数。
For x − 5 = 9, add 5 to both sides: x = 14. For 4x = 28, divide both sides by 4: x = 7.
例如 x − 5 = 9，两边同时加 5，得到 x = 14。对于 4x = 28，两边同时除以 4，得到 x = 7。
| Operation | Inverse operation |
|---|---|
| + a | − a |
| − a | + a |
| × a | ÷ a |
| ÷ a | × a |
Keeping a short table of inverse operations nearby is useful when you start. It prevents you from using the wrong operation under time pressure.
开始时在手边放一张简短的逆运算表很有用。它可以防止你在时间紧迫时用错运算。
4. Solving Two-Step Equations | 解两步方程
r subtraction first, then undo multiplication or division. For example, solve 2x + 5 = 17.
两步方程需要两次逆运算。标准的顺序是先去加减，再去乘除。例如，解方程 2x + 5 = 17。
both sides: 2x = 12. Then divide both sides by 2: x = 6. Always record each step clearly.
首先两边同时减去 5：2x = 12。然后两边同时除以 2：x = 6。务必清晰记录每一步。
2x + 5 = 17
2x = 17 − 5
2x = 12
x = 6
but this usually introduces fractions: x + 2.5 = 8.5. The balance method keeps it simpler.
如果你愿意，也可以先把每一项都除以 2，但这通常会引入分数：x + 2.5 = 8.5。平衡法会让计算更简单。
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