📚 Student’s Edition Grade 1 Lesson 16-2: Solving Linear Equations | 学生版 一年级 第16课 第2部分：解一元一次方程
In this lesson, we build on your earlier work with expressions and simple equations to solve linear equations systematically. A linear equation is an equation where the unknown, usually written as x, appears only to the first power. Mastering these techniques is essential for IGCSE Mathematics because nearly every algebra topic depends on rearranging and solving equations confidently.
在本课中，我们将在你之前学习代数式和简单方程的基础上，系统地解一元一次方程。一元一次方程是指未知数（通常写作 x）只出现一次方的方程。掌握这些技巧对 IGCSE 数学至关重要，因为几乎所有代数主题都依赖于熟练地变形和求解方程。
1. What is a Linear Equation? | 什么是一元一次方程？
A linear equation in one variable is any equation that can be written in the form ax + b = c, where a, b, and c are numbers, a is not zero, and x is the unknown. Examples include 2x + 3 = 11, 5x – 7 = 18, and x/4 + 1 = 3. The word linear means the variable appears only to the power 1, so there are no terms like x², x³, or 1/x.
一元一次方程是任何可以写成 ax + b = c 形式的方程，其中 a、b、c 是数，a 不为零，x 是未知数。例如 2x + 3 = 11、5x – 7 = 18 和 x/4 + 1 = 3。线性一词表示变量只出现一次方，因此没有 x²、x³ 或 1/x 这样的项。
2x + 3 = 11 5x – 7 = 18 x/4 + 1 = 3
2. The Balance Method | 天平法（等式性质）
Think of an equation as a balance scale. The equals sign is the pivot, and both sides must always have the same value. Whatever operation you perform on one side, you must also perform on the other side. This is the golden rule: keep the equation balanced. For example, if you add 4 to the left side, you must add 4 to the right side. If you divide the left side by 2, you must divide the right side by 2. This rule allows you to isolate the unknown step by step.
把方程想象成一架天平。等号是支点，两边必须始终保持相同的值。无论你对一边进行什么运算，也必须对另一边进行同样的运算。这是黄金法则：保持方程平衡。例如，如果你在左边加 4，就必须在右边加 4。如果你把左边除以 2，就必须把右边也除以 2。这条法则使你能够一步一步地把未知数分离出来。
3. Solving by Adding or Subtracting | 通过加减解方程
When a number is added to or subtracted from the unknown, use the inverse operation to remove it. For example, to solve x + 6 = 15, subtract 6 from both sides: x + 6 – 6 = 15 – 6, so x = 9. To solve x – 9 = 20, add 9 to both sides: x – 9 + 9 = 20 + 9, so x = 29. Always check by substituting your answer back into the original equation.
当一个数加到未知数上或从未知数中减去时，使用逆运算将其消去。例如，解 x + 6 = 15，两边同时减去 6：x + 6 – 6 = 15 – 6，所以 x = 9。解 x – 9 = 20，两边同时加 9：x – 9 + 9 = 20 + 9，所以 x = 29。务必把你的答案代回原方程进行检验。
x + 6 = 15 → x = 9
x – 9 = 20 → x = 29
4. Solving by Multiplying or Dividing | 通过乘除解方程
If the unknown is multiplied by a number, divide both sides by that number. If the unknown is divided by a number, multiply both sides by that number. For example, solve 4x = 28: divide both sides by 4, giving x = 7. Solve x/5 = 3: multiply both sides by 5, giving x = 15. This uses the inverse relationship between multiplication and division.
如果未知数乘以一个数，就把两边都除以这个数。如果未知数除以一个数，就把两边都乘以这个数。例如，解 4x = 28：两边都除以 4，得到 x = 7。解 x/5 = 3：两边都乘以 5，得到 x = 15。这利用了乘法和除法之间的逆运算关系。
4x = 28 → x = 7
x/5 = 3 → x = 15
5. Two-Step Equations | 两步方程
Many IGCSE equations require two steps. For example, solve 2x + 3 = 11. First subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4. The order is important: undo addition or subtraction before multiplication or division. This is the reverse of the order of operations.
许多 IGCSE 方程需要两步才能解出。例如，解 2x + 3 = 11。首先两边同时减去 3：2x = 8。然后两边同时除以 2：x = 4。顺序很重要：先消去加法或减法，再消去乘法或除法。这与运算顺序相反。
2x + 3 = 11 → 2x = 8 → x = 4
6. Equations with Brackets | 含括号的方程
When an equation contains brackets, expand them first using the distributive property. For example, solve 3(x – 2) = 12. Expand the left side: 3x – 6 = 12. Then add 6 to both sides: 3x = 18. Finally divide by 3: x = 6. Alternatively, you can divide both sides by 3 first: x – 2 = 4, then add 2: x = 6. Both methods give the same answer.
当方程含有括号时，先用分配律展开。例如，解 3(x – 2) = 12。展开左边：3x – 6 = 12。然后两边加 6：3x = 18。最后除以 3：x = 6。或者，你可以先两边除以 3：x – 2 = 4，然后加 2：x = 6。两种方法得到相同的答案。
3(x – 2) = 12 → 3x – 6 = 12 → x = 6
7. Equations with Unknowns on Both Sides | 未知数在等号两边的方程
If the unknown appears on both sides of the equals sign, collect the x terms on one side and the numbers on the other. For example, solve 5x + 2 = 2x + 14. Subtract 2x from both sides: 3x + 2 = 14. Then subtract 2 from both sides: 3x = 12. Finally divide by 3: x = 4. Always choose the side that keeps the coefficient of x positive to reduce errors.
如果未知数出现在等号的两边，就把含 x 的项移到一边，把数字移到另一边。例如，解 5x + 2 = 2x + 14。两边同时减去 2x：3x + 2 = 14。然后两边同时减去 2：3x = 12。最后除以 3：x = 4。一般选择使 x 的系数保持为正的一边，以减少错误。
5x + 2 = 2x + 14 → 3x = 12 → x = 4
8. Equations with Fractions | 含分数的方程
To solve equations involving fractions, multiply every term by the lowest common denominator to clear the fractions. For example, solve x/3 + 1 = 5/6. The lowest common denominator of 3 and 6 is 6. Multiply each term by 6: 6(x/3) + 6(1) = 6(5/6), which gives 2x + 6 = 5. Then subtract 6: 2x = -1. Divide by 2: x = -1/2 or -0.5. Always check that your solution does not make any denominator zero.
解含分数的方程时，将每一项都乘以最小公分母，以消去分数。例如，解 x/3 + 1 = 5/6。3 和 6 的最小公分母是 6。把每一项乘以 6：6(x/3) + 6(1) = 6(5/6)，得到 2x + 6 = 5。然后减 6：2x = -1。除以 2：x = -1/2 或 -0.5。一定要检验你的解不会使任何分母为零。
x/3 + 1 = 5/6 → 2x + 6 = 5 → x = -1/2
9. Forming Equations from Word Problems | 由应用题建立方程
Many exam questions ask you to write and solve an equation from a real-life situation. For example: “Three more than twice a number is 19. Find the number.” Let the number be x. Twice the number is 2x, and three more is 2x + 3. The equation is 2x + 3 = 19. Subtract 3: 2x = 16. Divide by 2: x = 8. The number is 8. Always define your variable clearly.
许多考试题目要求你根据实际问题列出并解方程。例如：”一个数的两倍再加 3 等于 19。求这个数。” 设这个数为 x。这个数的两倍是 2x，再加 3 是 2x + 3。方程为 2x + 3 = 19。减 3：2x = 16。除以 2：x = 8。这个数是 8。一定要清楚地定义你的变量。
2x + 3 = 19 → x = 8
10. Common Mistakes and Checking Answers | 常见错误与检验答案
A common mistake is to perform an operation on only one side of the equation. Remember the balance rule: what you do to one side, you must do to the other. Another mistake is to incorrectly apply the distributive property, for example writing 3(x – 2) = 3x – 2 instead of 3x – 6. To check your answer, substitute it back into the original equation. If both sides are equal, your solution is correct.
一个常见错误是只对方程的一边进行运算。记住天平法则：你对一边做什么，就必须对另一边做同样的事。另一个错误是错误地应用分配律，例如把 3(x – 2) 写成 3x – 2 而不是 3x – 6。要检验答案，把它代回原方程。如果两边相等，你的解就是正确的。
For example, if you solve 2x + 3 = 11 and get x = 4, substitute: 2(4) + 3 = 8 + 3 = 11, which matches the right side. This confirms that x = 4 is correct.
例如，如果你解 2x + 3 = 11 得到 x = 4，代入：2(4) + 3 = 8 + 3 = 11，与右边相等。这证明 x = 4 是正确的。
11. Summary and Practice Tips | 总结与练习建议
To master linear equations, practice the following steps: simplify both sides if needed, expand brackets, collect like terms, move variable terms to one side and numbers to the other, then isolate the variable using inverse operations. Check your solution by substitution. Use past IGCSE papers and timed practice to build speed and accuracy.
要掌握一元一次方程，请练习以下步骤：如有需要先化简两边，展开括号，合并同类项，把变量项移到一边、数字移到另一边，然后用逆运算分离变量。通过代入检验你的解。使用 IGCSE 历年真题和限时练习来提高速度和准确率。
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