Solving One-Step Equations | 解一元一次方程

📚 Solving One-Step Equations | 解一元一次方程

Equations are the foundation of algebra, and learning to solve them is one of the most important skills in KS3 mathematics. In this article, we will explore one-step equations — the simplest type of equation where only one operation is needed to find the unknown value. By mastering these, you will build the confidence to tackle more complex problems later. We will cover the balance method, practical solving techniques for each operation, and common mistakes to avoid.

方程是代数的基础，学习解方程是中学关键阶段三（KS3）数学中最重要的技能之一。本文我们将探索一步方程 —— 这是最简单的一类方程，只需一步运算就能求出未知数的值。掌握一步方程后，你将更有信心挑战后续更复杂的问题。我们将学习天平法、针对每种运算的实用解法，以及需要避免的常见错误。

1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal, using the equals sign ‘ = ‘. For example, x + 3 = 7 is an equation. It tells us that when we add 3 to the unknown number x, the result is 7. The letter x is called the variable, and the number 7 is the constant. Equations are like puzzles — our job is to find the value that makes the left side exactly equal to the right side.

方程是表示两个表达式相等的一种数学陈述，其中使用了等号 “ = ”。例如，x + 3 = 7 就是一个方程。它告诉我们，当未知数 x 加上 3 时，结果是 7。字母 x 被称为变量，数字 7 是常数。方程就像谜题 —— 我们的任务是找出让左边恰好等于右边的那个数值。

The goal in solving an equation is to isolate the variable on one side of the equals sign. For x + 3 = 7, we need to get x by itself. To do that, we must ‘undo’ whatever operation is being done to the variable. This is where the balance method comes in.

解方程的目标是把变量孤立在等号的一边。对于 x + 3 = 7，我们需要让 x 单独留在左边。为此，我们必须 “撤销” 对变量进行的任何运算。这时天平法就派上用场了。

---

2. The Balance Method | 天平法

Think of an equation as a pair of scales in perfect balance: the left side and the right side have exactly the same weight. If you add or remove something from one side, the scales will tip — unless you do exactly the same to the other side. The balance method is simply this: whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced and the equals sign true.

把方程想象成一副完美平衡的天平：左边和右边的重量完全相等。如果你在一边添加或移除一些东西，天平就会倾斜 —— 除非你对另一边也做完全相同的操作。天平法简单来说就是：你对方程的一边做了什么，就必须对另一边做同样的事。这样才能保持方程平衡，等号仍然成立。

For example, in x + 5 = 12, to remove the +5, we subtract 5 from the left side. To keep the balance, we must also subtract 5 from the right side. This gives us x = 7. The balance method is the golden rule of algebra and forms the basis for solving all linear equations.

例如，在 x + 5 = 12 中，为了去掉 +5，我们从左边减去 5。为了保持平衡，我们还必须从右边减去 5。这样我们就得到了 x = 7。天平法是代数的黄金法则，也是解所有线性方程的基础。

---

3. Solving Equations with Addition | 解含加法的方程

When a number is added to the variable, we undo the addition by performing the inverse operation — subtraction. The aim is to cancel out the number on the variable’s side.

当一个数字被加到变量上时，我们通过逆运算 —— 减法 来撤销加法。目标是把变量一侧的数字抵消掉。

Example: Solve x + 9 = 15.

例题：解 x + 9 = 15。

Step 1: Subtract 9 from both sides.
x + 9 − 9 = 15 − 9

步骤1：两边同时减去 9。
x + 9 − 9 = 15 − 9

Step 2: Simplify.
x = 6

步骤2：化简。
x = 6

Check: 6 + 9 = 15 ✓. Always verify by substituting your answer back into the original equation. The balance method ensures the solution is correct.

检验：6 + 9 = 15 ✓。总是通过把答案代回原方程来验证。天平法保证了解的正确性。

---

4. Solving Equations with Subtraction | 解含减法的方程

If a number is being subtracted from the variable, we use addition to cancel it out. Addition is the inverse of subtraction.

如果一个数字从变量中被减去，我们使用加法来抵消它。加法是减法的逆运算。

Example: Solve x − 4 = 11.

例题：解 x − 4 = 11。

Add 4 to both sides:
x − 4 + 4 = 11 + 4

两边同时加上 4：
x − 4 + 4 = 11 + 4

Simplify:
x = 15

化简：
x = 15

Check: 15 − 4 = 11 ✓. It is crucial to remember that the variable is still positive. Many students mistakenly write x = 11 − 4 when they see subtraction; always apply the inverse operation to both sides instead of moving terms across.

检验：15 − 4 = 11 ✓。记住变量仍然是正数至关重要。很多学生看到减法会错误地写成 x = 11 − 4；一定要对两边同时运用逆运算，而不是简单地把项移过去。

---

5. Solving Equations with Multiplication | 解含乘法的方程

When the variable is multiplied by a number, we undo the multiplication by dividing both sides by that same number. The coefficient (the number in front of x) tells us how many times x has been multiplied. For example, in 3x = 21, the coefficient is 3, meaning x has been multiplied by 3.

当变量乘以一个数字时，我们通过把两边同时除以这个数来撤销乘法。系数（x 前面的数字）告诉我们 x 被乘了多少倍。例如，在 3x = 21 中，系数是 3，意味着 x 被乘了 3 倍。

Example: Solve 5x = 35.

例题：解 5x = 35。

Divide both sides by 5:
5x ÷ 5 = 35 ÷ 5

两边同时除以 5：
5x ÷ 5 = 35 ÷ 5

Simplify:
x = 7

化简：
x = 7

Check: 5 × 7 = 35 ✓. Division is the inverse of multiplication, so this step always works as long as we perform it on both sides.

检验：5 × 7 = 35 ✓。除法是乘法的逆运算，所以只要两边都这么操作，这一步永远有效。

---

6. Solving Equations with Division | 解含除法的方程

If the variable is divided by a number, we use multiplication to isolate the variable. The equation might be written as x/4 = 6 or x ÷ 4 = 6. Both mean the same thing: x split into 4 equal parts gives 6.

如果变量被一个数除，我们使用乘法来隔离变量。方程可能写成 x/4 = 6 或 x ÷ 4 = 6。二者的意思相同：x 被分成 4 等份后，每份是 6。

Example: Solve x/3 = 9.

例题：解 x/3 = 9。

Multiply both sides by 3:
(x/3) × 3 = 9 × 3

两边同时乘以 3：
(x/3) × 3 = 9 × 3

Simplify:
x = 27

化简：
x = 27

Check: 27 ÷ 3 = 9 ✓. Using fractions is very common in one-step equations, and you must be comfortable converting between division and multiplication. Never leave a variable divided by a number; multiplication clears the denominator straight away.

检验：27 ÷ 3 = 9 ✓。分数在一步方程中非常常见，你必须熟练在除法和乘法之间转换。绝不要让变量被一个数除着而不管；乘法可以直接把分母去掉。

---

7. Mixed One-Step Equations | 混合一步方程

In practice, you will encounter a mix of all four operations. The key is to identify the operation being applied to the variable and then perform the opposite operation to both sides. The table below summarizes the steps.

在实际中，你会遇到四种运算混合出现的情况。关键是识别出对变量施加的运算，然后对两边执行相反的运算。下表总结了具体步骤。

Operation on variable 对变量的运算	Inverse operation 逆运算	Example 示例
Addition (x + a = b)	Subtraction (subtract a from both sides) 减法（两边同时减去 a）	x + 2 = 10 → x = 8
Subtraction (x − a = b)	Addition (add a to both sides) 加法（两边同时加上 a）	x − 5 = 3 → x = 8
Multiplication (ax = b)	Division (divide both sides by a) 除法（两边同时除以 a）	4x = 20 → x = 5
Division (x/a = b)	Multiplication (multiply both sides by a) 乘法（两边同时乘以 a）	x/6 = 3 → x = 18

Practice with mixed exercises helps you build speed and accuracy. Write down each step clearly; do not try to do everything mentally, especially when starting out.

做混合练习有助于提高速度和准确性。清楚写下每一步；尤其是在刚开始时，不要试图全凭心算。

---

8. Checking Your Answer | 验证答案

Checking is non-negotiable. Substitute the value you have found back into the original equation. If the left side equals the right side, your answer is correct. If not, review your steps. For example, if you solved x/2 = 7 and got x = 14, check: 14 ÷ 2 = 7, so it is correct. If you had mistakenly said x = 3.5, checking would immediately show the error because 3.5 ÷ 2 = 1.75 ≠ 7.

验证是必须做的。把你找到的值代回原方程。如果左边等于右边，你的答案就是正确的。如果不相等，请重新检查解题步骤。例如，如果你解 x/2 = 7 得到 x = 14，检验：14 ÷ 2 = 7，所以正确。如果你错误地认为 x = 3.5，检验立刻会显示错误，因为 3.5 ÷ 2 = 1.75 ≠ 7。

Using a mental or written check also reinforces your understanding of the relationship between variables and constants. Many exam questions specifically ask you to check your solutions, so developing this habit now will save marks later.

通过心算或书面检验还能强化你对变量和常数之间关系的理解。很多考试题会明确要求你验证答案，所以现在养成这个习惯将来能避免丢分。

---

9. Common Mistakes to Avoid | 常见错误

Mistake 1: Forgetting to do the same thing to both sides. Some students subtract only from one side, leading to an unbalanced equation. Always apply the inverse operation to both sides equally.

错误1：忘了对两边做相同的操作。有些学生只从一边减去数字，导致方程失衡。始终对两边同样地应用逆运算。

Mistake 2: Incorrect inverse operation. If the variable has a number subtracted, adding the same number cancels it. Don’t subtract again by accident. For x − 3 = 8, the correct step is x − 3 + 3 = 8 + 3, giving x = 11. Doing x − 3 − 3 = 8 − 3 would give x − 6 = 5, which is wrong.

错误2：逆运算用错。如果变量减去了一个数，加上这个数就能抵消它。不要不小心又减去。对于 x − 3 = 8，正确的步骤是 x − 3 + 3 = 8 + 3，得到 x = 11。如果做 x − 3 − 3 = 8 − 3 就会得到 x − 6 = 5，这就错了。

Mistake 3: Dividing when you should multiply. In x/4 = 5, some students divide by 4 instead of multiplying by 4. Remember the inverse of division is multiplication. Multiplying by 4 gives x = 20, which is correct.

错误3：该乘的时候却除了。在 x/4 = 5 中，有些学生会除以 4 而不是乘以 4。记住，除法的逆运算是乘法。乘以 4 得到 x = 20，这才是对的。

Mistake 4: Sign errors. Negative numbers appear frequently. For x + (−2) = 6, you can rewrite as x − 2 = 6, then add 2 to both sides to get x = 8. Be careful with double negatives.

错误4：符号错误。负数经常出现。对于 x + (−2) = 6，你可以把它重写成 x − 2 = 6，然后两边加 2 得到 x = 8。小心双重负号。

---

10. Real-Life Applications | 实际应用

One-step equations model many everyday situations. For example, if a taxi charges a flat rate of £3 plus £2 per mile, and the total fare is £15, you can represent the miles travelled as m with the equation 2m + 3 = 15. Although this is a two-step equation, the concept of isolating the variable begins with one-step thinking. Simply put, if you know the total cost and want to find how many items you bought, you might set up an equation like 4x = 20. Solving it tells you the number of items.

一步方程可以模拟很多日常情境。例如，如果出租车起步价为 3 英镑，之后每英里 2 英镑，总车费为 15 英镑，你可以用方程 2m + 3 = 15 来表示行驶的英里数 m。虽然这是个两步方程，但隔离变量的概念始于一步方程的思路。简单地说，如果你知道总花费，想要知道买了多少件商品，你可能会列出方程 4x = 20。解出它就知道商品数量。

Another example is sharing: if 5 friends share a bill equally and each pays £7, the total bill is represented by b/5 = 7, giving b = 35. Understanding the inverse operations behind these scenarios helps you make sense of everyday arithmetic. Step by step, algebra becomes a tool for solving practical problems, not just an abstract exercise.

另一个例子是分摊：如果 5 个朋友平摊一笔账单，每人付 7 英镑，那么账单总金额可以由 b/5 = 7 表示，解得 b = 35。理解这些场景背后的逆运算有助于你弄懂日常的算术。一步步地，代数就变成了解决实际问题的工具，而不仅仅是抽象的练习。

---

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）