Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the foundation of algebra at KS3. Learning to solve them confidently helps with harder topics such as graphs, formulae and simultaneous equations.

一元一次方程是 KS3 代数的基础。熟练求解这些方程有助于学习更难的课题，如图像、公式和联立方程。

1. What Is a Linear Equation? | 什么是一元一次方程

ion is a mathematical statement that two expressions are equal. It contains an equals sign '='. A linear equation in one variable has the variable raised only to the power 1.

方程是表示两个表达式相等的数学陈述，其中包含等号 '='。一元一次方程中的未知数只出现一次幂，即最高次数为 1。

re not linear because the variable has a power other than 1 or appears in the denominator.

例如，2x + 3 = 11 和 5y − 4 = 16 是一元一次方程。而 x² = 9 或 1/x = 2 不是一元一次方程，因为未知数的次数不是 1 或出现在分母中。

When solving a linear equation, the goal is to find the value of the variable that makes the statement true.

解一元一次方程的目标是找出使等式成立的未知数的值。

2. The Balance Method | 天平法（等式平衡）

Think of an equation as a balanced set of scales. Both sides have the same value. If you add, subtract, multiply or divide one side, you must do exactly the same to the other side to keep the equation balanced.

把方程想象成一台平衡的天平，两边质量相等。如果你对一边进行加、减、乘或除，必须对另一边做完全相同的运算，才能保持等式平衡。

For example, to solve x + 5 = 12, subtract 5 from both sides:

例如，要解 x + 5 = 12，在两边同时减去 5：

x + 5 − 5 = 12 − 5 → x = 7

This method works for all linear equations and helps you avoid mistakes when steps become more complicated.

这个方法适用于所有一元一次方程，并且当步骤变得更复杂时，它能帮助你避免出错。

3. Using Inverse Operations | 使用逆运算求解

Inverse operations reverse each other: addition and subtraction are inverses, multiplication and division are inverses. To isolate the variable, apply inverse operations in the reverse order of the usual order of operations.

逆运算互为相反：加法与减法互为逆运算，乘法与除法互为逆运算。要分离未知数，应按照通常运算顺序的相反顺序应用逆运算。

For the equation 3x + 4 = 19, first subtract 4 and then divide by 3:

对于方程 3x + 4 = 19，先减去 4，再除以 3：

3x + 4 − 4 = 19 − 4 → 3x = 15 → x = 5

The key is to undo what has been done to the variable step by step.

关键是要一步一步地撤销对未知数所做的运算。

4. Solving Equations with Brackets | 解带括号的方程

When an equation contains brackets, you can either expand the brackets first or divide both sides by a common factor if possible.

当方程中含有括号时，你可以先展开括号，或者在可能的情况下将两边同时除以一个公因数。

Consider 2(x + 3) = 14. Expanding the left side gives:

考虑方程 2(x + 3) = 14。展开左边得到：

2x + 6 = 14 → 2x = 8 → x = 4

Alternatively, divide both sides by 2 first:

另一种方法是先将两边除以 2：

2(x + 3) ÷ 2 = 14 ÷ 2 → x + 3 = 7 → x = 4

Both approaches give the same answer, so choose the one that makes the arithmetic easier.

两种方法得到相同的答案，因此选择计算更简单的一种即可。

5. Equations with the Variable on Both Sides | 两边都含未知数的方程

When the variable appears on both sides of the equation, collect all the variable terms on one side and all the constant terms on the other side.

当未知数出现在方程两边时，把含未知数的项移到一边，把常数项移到另一边。

For example, to solve 5x + 2 = 2x + 11:

例如，要解 5x + 2 = 2x + 11：

5x − 2x = 11 − 2 → 3x = 9 → x = 3

Remember that when you move a term across the equals sign, you must change its sign to its opposite.

记住，当你把一项移到等号另一边时，必须把它的符号改为相反的符号。

Check by substituting x = 3: left side is 5(3) + 2 = 17 and right side is 2(3) + 11 = 17.

将 x = 3 代入检验：左边为 5(3) + 2 = 17，右边为 2(3) + 11 = 17。

6. Equations Involving Fractions | 含分数的方程

To solve an equation that contains fractions, multiply every term on both sides by the lowest common multiple (LCM) of all denominators. This clears the fractions and gives a simpler equation.

解含有分数的方程时，将两边每一项都乘以所有分母的最小公倍数（LCM）。这样可以消去分数，得到更简单的方程。

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For x/4 + 1 = 3, multiply every term by 4:

对于 x/4 + 1 = 3，将每一项乘以 4：

4 × (x/4) + 4 × 1 = 4 × 3 → x + 4 = 12 → x = 8

For (2x)/3 = 6, multiply both sides by 3 and then divide by 2:

对于 (2x)/3 = 6，两边先乘以 3，再除以 2：

3 × (2x)/3 = 3 × 6 → 2x = 18 → x = 9

Always write the multiplication for every term, so you do not forget a constant term or a denominator.

务必将每一项都乘以分母的最小公倍数，这样就不会漏掉常数项或分母。

7. Forming Equations from Word Problems | 从实际问题建立方程

Word problems require you to translate a written statement into an algebraic equation before solving it. Follow these steps: read carefully, choose a variable, write the equation, solve it, and then answer the question in words.

应用题要求你先将文字叙述转化为代数方程，然后再求解。可以按照以下步骤：仔细读题，设未知数，写出方程，求解，最后用文字回答问题。

Example: Three times a number plus 5 equals 20. Let the number be n.

例题：一个数的 3 倍加 5 等于 20。设这个数为 n。

3n + 5 = 20 → 3n = 15 → n = 5

The number is 5. Always check that your answer makes sense in the original context.

这个数是 5。一定要检查答案在原题情境中是否合理。

Common phrases include: 'more than' means addition, 'less than' means subtraction, 'times' means multiplication, and 'shared between' means division.

常见短语包括：'more than' 表示加法，'less than' 表示减法，'times' 表示乘法，'shared between' 表示除法。

8. Checking Your Solution | 检验答案

After solving an equation, substitute your answer back into the original equation to make sure both sides are equal.

解完方程后，将答案代入原方程，确认两边相等。

For 2(x − 3) = 10, the solution is x = 8.

对于方程 2(x − 3) = 10，解为 x = 8。

Left side = 2(8 − 3) = 2 × 5 = 10, which equals the right side.

左边 = 2(8 − 3) = 2 × 5 = 10，与右边相等。

Substitution is the fastest way to catch arithmetic or sign errors, especially in tests.

代入检验是发现算术错误或符号错误的最快方法，尤其是在考试中。

9. Common Mistakes to Avoid | 常见错误与避免方法

Many students lose marks on simple errors. Watch out for these common mistakes.

许多学生会因为一些简单错误而丢分。请注意以下常见错误。

• Forgetting to do the same operation to both sides. 忘记对等号两边进行相同的运算。
• Incorrectly changing signs when moving terms. 移项时符号改变错误。
• Expanding brackets incorrectly, such as writing 2(x + 3) = 2x + 3 instead of 2x + 6. 括号展开错误，例如把 2(x + 3) 写成 2x + 3 而不是 2x + 6。
• in the wrong order, for example solving 3x + 4 = 19 by dividing by 3 before subtracting 4. 运算顺序错误，例如解 3x + 4 = 19 时先除以 3 而不是先减去 4。

Taking a moment to check each step can prevent these errors and improve accuracy.

花一点时间检查每一步可以防止这些错误并提高准确性。

10. Key Practice Questions | 关键练习题

Try these practice questions to build speed and confidence. Cover the answers and solve each equation step by step.

尝试以下练习题，提高速度和信心。遮住答案，一步一步地解每个方程。

Equation 方程	Solution 解
x + 9 = 15	x = 6
4x = 28	x = 7
3x − 5 = 10	x = 5
2(x + 4) = 18	x = 5
6x + 2 = 4x + 12	x = 5

If you can solve these quickly and check them by substitution, you are well prepared for KS3 assessments.

如果你能快速解出这些方程并通过代入检验，那么你已经为 KS3 评估做好了充分准备。

11. Summary and Exam Tips | 总结与应试技巧

Always show clear working, because method marks are often awarded even if the final answer is wrong.

始终写出清晰的解题步骤，因为即使最终答案错误，过程分往往也会给出。

Use the balance method and inverse operations consistently. In an exam, check your solution by substitution when time allows.

始终使用天平法和逆运算。考试中如果时间允许，用代入法检验答案。

Read word problems twice and define the variable clearly before forming the equation.

将应用题读两遍，在列方程之前清楚定义未知数。

Finally, keep practising a variety of equations, including those with brackets, fractions and variables on both sides.

最后，持续练习各种类型的方程，包括含括号、含分数和两边都含未知数的方程。

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