Solving Linear Equations with Unknowns on Both Sides | 含双边未知数的一元一次方程求解

📚 Solving Linear Equations with Unknowns on Both Sides | 含双边未知数的一元一次方程求解

When you first learned to solve equations, the unknown usually appeared on only one side of the equals sign. As you progress through KS3 Cambridge Mathematics, you encounter equations where the unknown appears on both sides, such as 3x + 1 = 2x + 5. Mastering these equations is an essential skill because it deepens your understanding of algebraic manipulation and prepares you for more advanced topics like simultaneous equations and functions. In this article, we will build your confidence step by step, using clear examples and the balancing method, so that you can solve any linear equation with unknowns on both sides accurately and efficiently.

当你刚开始学习解方程时，未知数通常只出现在等号的一侧。随着你在KS3剑桥数学课程中不断进步，你会遇到未知数同时出现在两边的情况，例如 3x + 1 = 2x + 5。掌握这类方程是一项关键技能，因为它能加深你对代数变换的理解，并为学习联立方程和函数等更高级的课题打好基础。在本文中，我们将通过清晰的例题和平衡法一步一步建立你的信心，让你能够准确、高效地求解任何含有双边未知数的一元一次方程。

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1. Understanding the Basics | 理解基本概念

A linear equation is a mathematical statement that shows two expressions are equal, and it contains a variable (usually x) with no exponents greater than 1. When the unknown appears on both sides of the equals sign, there is an extra step required: you must collect the variable terms on one side and the constant terms on the other. For example, in the equation 4x − 3 = x + 6, both sides contain an x term. Your goal is to find the single value of x that makes the equality true. This type of equation often models real-life situations where two quantities depend on the same unknown and are set equal to each other.

一元一次方程是表明两个表达式相等并且包含一个变量（通常为x）的数学陈述，变量的指数不超过1。当未知数在等号两边都出现时，就需要一个额外的步骤：你必须把所有含未知数的项集中到等式的一边，常数项集中到另一边。例如，在方程 4x − 3 = x + 6 中，两边都有x项。你的目标是找到使等式成立的唯一的x值。这类方程通常可以模拟现实生活中的情形，即两个量都依赖于同一个未知数，并且被设定为彼此相等。

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2. The Balancing Method | 天平平衡法

Think of an equation as a perfectly balanced set of scales. Whatever you do to one side, you must do exactly the same to the other side to maintain the balance. If you add a weight to the left-hand pan, you must add the same weight to the right-hand pan. In algebra, this means you can add, subtract, multiply or divide both sides by the same non-zero quantity without changing the solution. When an equation has unknowns on both sides, the balancing method helps you eliminate the variable from one side by performing the same operation on both sides. This approach reduces the equation to the simpler form where the unknown is on only one side.

可以把方程想象成一个完全平衡的天平。无论你对一边做什么，都必须对另一边做完全相同的操作，才能保持平衡。如果你往左边托盘加一个砝码，就必须往右边托盘也加一个同样的砝码。在代数中，这意味着你可以对等式两边同时加、减、乘或除以同一个非零的量，而不会改变方程的解。当方程两边都含有未知数时，天平平衡法可以帮助你通过对两边执行相同的运算，从其中一边消去未知数。这样就把方程约简为未知数只在一侧的简单形式。

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3. Transposing Terms | 移项

Transposing, or moving terms from one side to the other, is a shortcut derived from the balancing method. To remove a term, you perform the inverse operation on both sides. For instance, to move “+ x” you subtract x from both sides; to move “− 5” you add 5 to both sides. Many students find it helpful to visualise ‘undoing’ a term. A common rule is: when a term changes side, its sign changes. So 3x + 2 = 5x becomes 3x − 5x = −2 after subtracting 5x from both sides and subtracting 2 from both sides in one go. However, always remember that you are actually using the balancing method; the sign-change rule is just a quick mental step, but it must be applied correctly, especially with negative coefficients.

移项，也就是把项从等式的一边移动到另一边，是从平衡法推导出来的一种快捷方法。要消去一项，你可以在两边同时进行逆运算。例如，要移走”+ x”，你就在两边同时减去x；要移走”− 5″，就在两边同时加上5。许多学生发现想象“撤销”某一项很有帮助。一个常见的规则是：当一项改变边时，它的符号要改变。因此 3x + 2 = 5x 可以变为 3x − 5x = −2，即先同时减去5x，再同时减去2，一步完成。不过，要始终记住你实际上是在运用天平平衡法；变号规则只是一条快速心算的捷径，必须正确使用，尤其是在处理负系数的时候。

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4. Step-by-Step Example 1 | 逐步示例 1

Let us solve 5x + 2 = 3x + 10.

我们来解方程 5x + 2 = 3x + 10。

Step 1: We want all x terms on the left. Subtract 3x from both sides.

步骤1：我们希望所有x项都在左边。两边同时减去3x。

5x + 2 − 3x = 3x + 10 − 3x

This simplifies to:

化简得：

2x + 2 = 10

Step 2: Now remove the constant “+2” from the left by subtracting 2 from both sides.

步骤2：现在通过两边同时减去2来消去左边的常数”+2″。

2x + 2 − 2 = 10 − 2

2x = 8

Step 3: Divide both sides by 2 to find x.

步骤3：两边同时除以2，求出x。

2x ÷ 2 = 8 ÷ 2

x = 4

We have found the solution: x = 4. You can verify it mentally: 5(4) + 2 = 22, and 3(4) + 10 = 22.

我们求出了解：x = 4。你可以心算验证：5(4) + 2 = 22，3(4) + 10 = 22。

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5. Step-by-Step Example 2 | 逐步示例 2

Solve 4x − 7 = x + 8.

解方程 4x − 7 = x + 8。

Here the unknown appears on both sides, and there is a negative constant. Subtract x from both sides to gather x terms on the left.

这里未知数出现在两边，而且有一个负常数。两边同时减去x，把x项集中到左边。

4x − 7 − x = x + 8 − x

3x − 7 = 8

Now add 7 to both sides to isolate the term with x.

现在两边同时加上7，把含有x的项单独留在左边。

3x − 7 + 7 = 8 + 7

3x = 15

Finally, divide both sides by 3.

最后，两边同时除以3。

3x ÷ 3 = 15 ÷ 3

x = 5

Check: 4(5) − 7 = 13, and 5 + 8 = 13. It is correct. Notice how the balancing method handles the negative sign smoothly.

检验：4(5) − 7 = 13，5 + 8 = 13，正确。注意天平平衡法是如何顺畅地处理负号的。

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6. Handling Fractions | 处理分数

Fractions in equations can look tricky, but the balancing method works just as well. Consider the equation x/2 + 3 = x − 1/2. The first strategy is often to eliminate fractions by multiplying every term on both sides by the common denominator, which is 2 in this case. Multiply both sides by 2:

方程里出现分数可能看起来棘手，但平衡法同样适用。考虑方程 x/2 + 3 = x − 1/2。第一个策略通常是用公分母乘两边的每一项来消去分母，这里的公分母是2。两边同时乘以2：

2 × (x/2 + 3) = 2 × (x − 1/2)

Distribute carefully: 2 × (x/2) = x, 2 × 3 = 6, so the left side becomes x + 6. On the right: 2 × x = 2x, and 2 × (−1/2) = −1, giving 2x − 1. The new equation is x + 6 = 2x − 1. Now collect x terms: subtract x from both sides to get 6 = x − 1. Then add 1: 7 = x. So x = 7. Check: 7/2 + 3 = 3.5 + 3 = 6.5, and 7 − 0.5 = 6.5. Always multiply every term, including constants, by the common denominator.

仔细分配：2 × (x/2) = x，2 × 3 = 6，因此左边变为 x + 6。右边：2 × x = 2x，2 × (−1/2) = −1，得出 2x − 1。新方程为 x + 6 = 2x − 1。现在集中x项：两边同时减去x，得到 6 = x − 1。再加1：7 = x，所以 x = 7。检验：7/2 + 3 = 3.5 + 3 = 6.5，7 − 0.5 = 6.5。务必把每一项，包括常数，都乘以公分母。

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7. Dealing with Negative Coefficients | 处理负系数

When the unknown has a negative coefficient, such as in −2x + 5 = x − 7, you can use the same strategy. One approach is to add 2x to both sides to make the coefficient positive on the right side.

当未知数的系数为负时，例如 −2x + 5 = x − 7，你可以使用相同的策略。一种方法是两边同时加上2x，使右边的系数变为正。

−2x + 5 + 2x = x − 7 + 2x

5 = 3x − 7

Now add 7 to both sides: 12 = 3x, so x = 4. Alternatively, you could subtract x from both sides to get −3x + 5 = −7, then subtract 5: −3x = −12, and divide by −3: x = 4. Both methods are valid. The key is to avoid sign errors; double-check your steps, especially when dividing by a negative number.

现在两边同时加上7：12 = 3x，因此 x = 4。另一种方法是两边同时减去x，得到 −3x + 5 = −7，再减去5：−3x = −12，再除以−3：x = 4。两种方法都有效。关键是要避免符号错误；复查每一步，特别是在除以负数的时候。

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8. Equations with Brackets First | 先展开括号的方程

If an equation contains brackets, always expand them before trying to collect like terms. For example, solve 2(x + 3) = 3x − 4.

如果方程含有括号，务必先展开括号再合并同类项。例如，解方程 2(x + 3) = 3x − 4。

2x + 6 = 3x − 4

Now we have a straightforward equation with unknowns on both sides. Subtract 2x from both sides:

现在我们得到了一个简单的双边未知数方程。两边同时减去2x：

6 = x − 4

Add 4 to both sides:

两边同时加4：

10 = x

Thus x = 10. Check: 2(10 + 3) = 26, and 3(10) − 4 = 26. Expanding correctly is vital; a common mistake is failing to multiply both terms inside the bracket by the factor outside.

因此 x = 10。检验：2(10 + 3) = 26，3(10) − 4 = 26。正确展开括号至关重要；一个常见错误是忘记把括号中的每一项都乘以外面的因数。

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9. Equations with Variables on Both Sides and Brackets | 同时有括号和双边变量

Let us tackle a more involved equation: 3(2x − 1) = 5(x + 2).

我们来解一道更复杂的方程：3(2x − 1) = 5(x + 2)。

First, expand both sides:

首先，展开两边：

6x − 3 = 5x + 10

Subtract 5x from both sides:

两边同时减去5x：

x − 3 = 10

Add 3 to both sides:

两边同时加3：

x = 13

Always perform the expansion before moving terms. If you try to move terms without expanding, you risk making errors. Check: 3(26 − 1) = 3 × 25 = 75, and 5(13 + 2) = 5 × 15 = 75.

一定要先展开再移项。如果试图不展开就移项，很可能出错。检验：3(26 − 1) = 3 × 25 = 75，5(13 + 2) = 5 × 15 = 75。

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10. Checking Your Solution | 检验解

Checking your answer is not just good practice; it can help you catch algebraic mistakes. Substitute your value back into the original equation, and evaluate each side separately. If both sides give the same number, your solution is correct. For an equation like 2x + 1 = 7 − x, suppose you found x = 2. Left side: 2(2) + 1 = 5. Right side: 7 − 2 = 5. Match. For more complex equations, do the arithmetic carefully, especially when negatives and fractions are involved. If the sides do not match, retrace your steps to find where a sign error or an arithmetic slip occurred.

检验答案不仅是好习惯，还能帮你发现代数运算中的错误。把你的解代入原方程，分别计算等号两边。如果两边得出相同的数值，你的解就是正确的。对于像 2x + 1 = 7 − x 这样的方程，假设你求出 x = 2。左边：2(2) + 1 = 5。右边：7 − 2 = 5，吻合。对于更复杂的方程，请仔细计算，特别是涉及负数和分数时。如果两边不相等，就回头检查步骤，找出在哪里发生了符号错误或计算失误。

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11. Common Mistakes to Avoid | 常见错误

Even strong students can slip up. Here are typical errors to watch for:

即使数学不错的学生也可能犯错。以下是一些需要警惕的典型错误：

• Forgetting to change the sign when transposing a term. Remember: moving a term across the equals sign changes its sign. For example, turning 3x + 2 = 5x into 3x − 5x = 2 is wrong; it should be −2 on the right.

  移项时忘记改变符号。记住：把一项移到等号另一边要改变它的符号。例如，把 3x + 2 = 5x 变为 3x − 5x = 2 是错误的；右边应该是 −2。

• Not expanding brackets correctly, especially distributing a negative sign, e.g., −2(x − 3) must become −2x + 6, not −2x − 6.

  没有正确展开括号，尤其是在分配负号时，例如，−2(x − 3) 必须变成 −2x + 6，而不是 −2x − 6。

• Applying an operation to only some terms. When multiplying both sides by a number, you must multiply every term, including constants.

  只对部分项进行运算。将等式两边同乘以一个数时，必须乘以每一项，包括常数。

• Ignoring the balance: adding or subtracting something on one side without doing it to the other.

  忽视平衡：在一边加减某个量而没有在另一边同样操作。

• Dividing by a negative incorrectly, which can flip the inequality if it were an inequality, but with equations, just be careful with signs.

  除以负数时出错，若是不等式会改变不等号方向，但对方程而言，只需注意符号即可。

Being aware of these pitfalls will help you develop careful and accurate solving habits.

意识到这些陷阱将帮助你养成仔细、准确解题的习惯。

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12. Practice Problems | 练习题

Try these equations to strengthen your understanding. Write out full steps using the balancing method, and check each solution.

尝试以下方程来加深理解。请使用平衡法写出完整步骤，并检验每个解。

• 7x − 5 = 3x + 11

  7x − 5 = 3x + 11

• 2(x + 4) = x + 10

  2(x + 4) = x + 10

• 3x/2 + 1 = x − 1/2

  3x/2 + 1 = x − 1/2

• 5 − 4x = 2x + 11

  5 − 4x = 2x + 11

• 2(3x − 1) = 4(x + 2)

  2(3x − 1) = 4(x + 2)

Work through each one systematically, and remember: if you ever feel stuck, return to the core principle – do the same thing to both sides to keep the scales balanced.

请逐题系统解答，记住：如果一时卡住，就回到核心原则——对两边做同样的事，保持天平平衡。

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