Solving One-Step Equations | 解一元一次方程

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

Welcome to the world of equations! In Key Stage 3 Mathematics, solving one-step equations is one of the first big steps into algebra. You will learn how to find the unknown value in a simple equation by using inverse operations. This skill is essential not only for your Cambridge Lower Secondary course but also for all the maths you’ll encounter later on. In this article, we’ll break down the balancing method, explore addition, subtraction, multiplication and division equations, and tackle word problems together. Let’s get started.

欢迎来到方程的世界！在 KS3 数学中，解一元一次方程是你进入代数学大门的重要第一步。你将学习如何利用逆运算，找出简单方程中的未知数。这项技能不仅对你的剑桥初中课程至关重要，也为你今后遇到的所有数学问题打下基础。本文将详细拆解天平平衡法，探讨涉及加减乘除的各种方程，并一起解决文字应用题。让我们开始吧。

1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal. It always contains an equals sign (=). For example, x + 5 = 12 is an equation. The letter x represents an unknown number, and our job is to find out what x must be so that both sides of the equals sign have the same value. In KS3, most equations you see will have one variable, like x or y, and the goal is to solve for that variable.

方程是一种数学陈述，表明两个表达式是相等的。它总是包含一个等号（=）。例如，x + 5 = 12 就是一个方程。字母 x 代表一个未知数，而我们要做的就是找出 x 的值，使得等号两边的数值相等。在 KS3 阶段，你见到的大多数方程都只有一个变量，例如 x 或 y，目标就是解出这个变量。

Equations are like a balance scale: the left side must equal the right side. Whatever we do to one side, we must do to the other to keep the equation balanced. This is the golden rule of algebra.

方程就像一架天平：左边必须等于右边。无论我们对一边做了什么操作，为了保持方程平衡，另一边也必须做同样的操作。这就是代数的黄金法则。

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2. Key Components of an Equation | 方程的关键组成部分

Before jumping into solving, let’s identify the parts of a one-step equation. Take 3x = 15. Here, 3 is the coefficient (the number multiplied by x), the variable is x, and the right-hand side is 15. In the equation y − 4 = 7, the constant term is −4, and the operation is subtraction. Recognising which operation connects the variable and the constant is crucial because you will apply the inverse operation to isolate the variable.

在动手解题之前，我们先来认识一下一元一次方程的各部分。以 3x = 15 为例，3 是系数（与 x 相乘的数），变量是 x，右边是 15。在方程 y − 4 = 7 中，常数项是 −4，运算方式是减法。识别出变量与常数之间是何种运算关系至关重要，因为你需要使用逆运算来隔离变量。

The variable is often called the unknown, and we usually use letters like x, y, or a. Equations can also have the variable on the right side, such as 20 = p + 8. Don’t get confused—the rules still apply the same way.

变量常被称为未知数，我们通常使用 x、y 或 a 等字母。方程中变量也可能出现在右边，比如 20 = p + 8。不要被迷惑——规则依然同样适用。

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3. Balancing the Equation | 方程平衡

The balancing method is the foundation of solving equations. Think of an equation as a pair of old-fashioned scales in perfect balance. If you add a weight to one side, you must add exactly the same weight to the other side to keep it balanced. This means: what you do to one side of an equation, you must do to the other.

天平平衡法是解方程的基石。可以把方程想象成一对完全平衡的老式天平。如果你在一侧加了一个砝码，必须在另一侧加上完全相同的砝码，才能保持平衡。这就意味着：你对方程的一边做了什么，就必须对另一边做相同的操作。

For instance, to solve x + 6 = 13, we want x by itself. The opposite (inverse) of adding 6 is subtracting 6. So we subtract 6 from both sides: x + 6 − 6 = 13 − 6, which simplifies to x = 7. The equation stays balanced because we performed the same operation on both sides.

例如，要解 x + 6 = 13，我们希望 x 单独在一边。加 6 的逆运算是减去 6。所以我们在两边同时减去 6：x + 6 − 6 = 13 − 6，化简得到 x = 7。方程保持平衡，因为我们对两边执行了相同的运算。

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4. Solving by Addition or Subtraction | 通过加法或减法求解

When a number is added to or subtracted from the variable, we use the opposite operation to cancel it. If the equation is x + 9 = 21, we subtract 9 from both sides to get x = 12. If the equation is y − 5 = 14, we add 5 to both sides: y − 5 + 5 = 14 + 5, so y = 19. This works every time because addition and subtraction are inverse operations.

当变量加上或减去一个数时，我们使用相反的运算来抵消它。如果方程是 x + 9 = 21，我们从两边减去 9，得到 x = 12。如果方程是 y − 5 = 14，我们在两边加上 5：y − 5 + 5 = 14 + 5，所以 y = 19。这总能行得通，因为加法和减法是一对逆运算。

Watch out for equations like 16 = a + 3. To solve, subtract 3 from both sides: 16 − 3 = a + 3 − 3, giving 13 = a or a = 13. The variable does not have to be on the left; the solution is the same.

小心像 16 = a + 3 这样的方程。要解它，两边同时减去 3：16 − 3 = a + 3 − 3，得到 13 = a 或 a = 13。变量不一定非要在左边；解是一样的。

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5. Solving by Multiplication or Division | 通过乘法或除法求解

When the variable is multiplied by a number, we divide both sides by that number. For 4x = 32, divide both sides by 4: (4x)/4 = 32/4, so x = 8. When the variable is divided by a number, such as y/6 = 5, we multiply both sides by 6: (y/6) × 6 = 5 × 6, giving y = 30. Multiplication and division are also inverse operations.

当变量乘上一个数时，我们两边除以这个数。对于 4x = 32，两边除以 4：(4x)/4 = 32/4，所以 x = 8。当变量除以一个数时，例如 y/6 = 5，两边乘以 6：(y/6) × 6 = 5 × 6，得到 y = 30。乘法和除法同样是一对逆运算。

Remember that the coefficient doesn’t have to be a whole number. For 0.5t = 12, you can divide by 0.5 to get t = 24, or multiply both sides by 2 if that feels easier. The key is keeping the equation balanced.

请记住，系数不一定是整数。对于 0.5t = 12，你可以除以 0.5 得到 t = 24，或者如果觉得更容易，两边乘以 2。关键是要保持方程平衡。

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6. One-Step Equations with Fractions | 含有分数的一元一次方程

Fractions in equations might look tricky, but the same inverse operations apply. Suppose you have (3/4)y = 9. To isolate y, you need to undo the multiplication by 3/4. The inverse is multiplying by the reciprocal 4/3. So y = 9 × (4/3), which equals 12. Another approach is to first multiply both sides by 4 to clear the denominator: 3y = 36, then divide by 3. Both paths lead to the correct answer.

方程里的分数看上去可能有些棘手，但同样的逆运算原则依然适用。假设你有 (3/4)y = 9。要隔离 y，你需要撤销乘以 3/4 的运算。其逆运算是乘以它的倒数 4/3。所以 y = 9 × (4/3)，等于 12。另一种方法是先两边乘以 4 消去分母：3y = 36，再除以 3。两种途径都能得到正确答案。

When the variable is in a fraction like x/2 = 7, it’s a division equation in disguise. Multiply both sides by 2 to get x = 14. Practice both methods to see which you prefer.

当变量处于分数形式中，比如 x/2 = 7，那其实是一个伪装成除法的方程。两边乘以 2 得到 x = 14。练习两种方法，看哪一种你更得心应手。

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7. Checking Your Solution | 检验解

Always check your answer by substituting it back into the original equation. If you solved 2x + 3 = 11 and got x = 4, replace x with 4: 2(4) + 3 = 8 + 3 = 11, which matches the right side. This confirms the solution is correct. In one-step equations, it’s a quick mental step that can save you from simple mistakes.

一定要把你的答案代回原方程进行检验。如果你解 2x + 3 = 11 得到 x = 4，就用 4 替换 x：2(4) + 3 = 8 + 3 = 11，与右边相符。这就证明解是正确的。在一元一次方程中，这是一个快速的心算步骤，可以让你避免简单的错误。

If checking shows the sides are not equal, go back and look for errors in your inverse operations or arithmetic. This habit builds strong algebra skills.

如果检验时发现两边不相等，就返回去检查逆运算或计算过程中的错误。这个习惯能帮你打下坚实的代数基础。

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8. Word Problems to Equations | 将文字题转化为方程

Many exam questions present equations as word problems. For example: ‘A number multiplied by 7 gives 56. What is the number?’ The equation is 7n = 56, so n = 8. Another example: ‘After spending £15, Alex has £27 left. How much did he start with?’ Let m be the starting amount: m − 15 = 27, so m = 42. The key is translating words into algebraic statements.

许多考试题目会以文字题的形式呈现方程。例如：“一个数乘以 7 得到 56。这个数是多少？”方程是 7n = 56，所以 n = 8。再比如：“亚历克斯花了 15 英镑后，还剩下 27 英镑。他最初有多少钱？”设最初金额为 m：m − 15 = 27，所以 m = 42。关键是把文字转换成代数语句。

Practise writing equations from scenarios like ‘half of a number is 20’ (x/2 = 20) or ‘four more than a number is 18’ (a + 4 = 18). As you practise, you’ll notice common patterns that make translation second nature.

多练习根据情景写出方程，比如“一个数的一半是 20”（x/2 = 20）或“一个数加四是 18”（a + 4 = 18）。通过练习，你会发现其中的常见模式，从而让转化成为你的第二天性。

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9. Common Mistakes to Avoid | 常见错误及避免方法

One common error is forgetting to perform the operation on both sides. For x − 3 = 10, a student might write x = 10 − 3 instead of x = 10 + 3. Always use inverse operations equally. Another mistake is misreading the equation, especially when the variable is on the right. Treat both sides symmetrically.

一个常见错误是忘了在等号两边同时进行运算。对于 x − 3 = 10，有的学生可能会写成 x = 10 − 3，而不是 x = 10 + 3。始终要公平地使用逆运算。另一个错误是误读方程，尤其是当变量在右边的时候。要对等号两侧一视同仁。

Also, be careful with negative coefficients. If you have −y = 7, multiply both sides by −1 to get y = −7. Don’t forget that the negative sign belongs to the variable. Double-checking your solution can catch these errors early.

此外，要小心负系数。如果你遇到 −y = 7，两边乘以 −1 得到 y = −7。别忘了负号是属于变量的。及早检查你的解可以抓出这些错误。

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10. Practice and Summary | 练习与总结

Now try these one-step equations to build confidence:

• x + 25 = 40
• y − 7 = 22
• 6a = 54
• b/4 = 12
• (2/3)c = 10

Solve each one using inverse operations, then check your answers. (Solutions: x = 15, y = 29, a = 9, b = 48, c = 15)

现在尝试这些一元一次方程来建立自信：

• x + 25 = 40
• y − 7 = 22
• 6a = 54
• b/4 = 12
• (2/3)c = 10

用逆运算解出每一个，然后检验答案。（解：x = 15, y = 29, a = 9, b = 48, c = 15）

To sum up, solving one-step equations is all about keeping the balance and using the opposite operation. Once you master this, you’ll be ready for two-step equations, brackets, and beyond. Keep practising, stay methodical, and algebra will soon feel as natural as breathing.

总而言之，解一元一次方程全在于保持平衡并使用逆运算。一旦你掌握了这一点，你就为学习两步方程、带括号的方程以及其他进阶内容做好了准备。不断练习，保持有条不紊，代数很快会像呼吸一样自然。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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