Solving Linear Equations with Unknowns on Both Sides | 求解两边含未知数的线性方程

📚 Solving Linear Equations with Unknowns on Both Sides | 求解两边含未知数的线性方程

nown x appears on both sides of an equation. This guide gives you a clear method for solving these equations confidently.

线性方程贯穿剑桥初中数学课程的始终。许多练习卷（包括编号为 p197_2 的练习题）都包含未知数 x 出现在等号两边的问题。本指南为你提供清晰的方法，帮助你自信地解出这些方程。

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1. Understanding Linear Equations | 理解线性方程

A linear equation is a statement that two expressions are equal. In KS3, the unknown is usually written as x or n. A linear equation has no powers such as x² or x³. Solving the equation means finding the value of x that makes the statement true.

线性方程是说明两个表达式相等的一种陈述。在 KS3 阶段，未知数通常写成 x 或 n。线性方程不含有 x² 或 x³ 这样的幂。解方程就是求出使等式成立的 x 的值。

r, but x² + 3 = 11 is not linear. Keep a careful eye on the highest power of the variable.

例如，2x + 3 = 11 是线性方程，但 x² + 3 = 11 不是线性方程。要仔细观察变量的最高次数。

2x + 3 = 11 is linear

x² + 3 = 11 is not linear

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2. The Balance Method | 天平法

Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side. This is called the balance method. You can add, subtract, multiply or divide, but you must apply the same operation to both sides.

把方程想象成一架平衡的天平。无论你对一边做了什么，都必须对另一边做同样的运算。这种方法称为天平法。你可以加、减、乘、除，但必须在方程的两边同时进行相同的运算。

m both sides to isolate x. The equation remains balanced, and you find the solution x = 7.

如果 x + 5 = 12，两边同时减去 5，使 x 单独留在一边。方程仍然保持平衡，你便得到解 x = 7。

x + 5 = 12

x + 5 − 5 = 12 − 5

x = 7

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3. Solving Two-Step Equations | 解两步方程

btract 3 from both sides, then divide both sides by 2. This reverse order follows the principle of undoing the operations around the unknown.

许多方程需要两个步骤。例如要解 2x + 3 = 11，首先两边减去 3，然后两边除以 2。这种逆序运算遵循了逐步消去未知数周围运算的原则。

Write each line of working clearly so an examiner can follow your method. Neatness is especially important when the question asks you to show all steps.

写出每一步清晰的解题过程，以便阅卷老师能够看懂你的方法。当题目要求展示所有步骤时，书写整洁尤为重要。

2x + 3 = 11

2x = 8

x = 4

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4. Expanding Brackets First | 先展开括号

out the bracket to get 2x + 6 = x + 8, then use the balance method to collect like terms.

当方程含有括号时，要先展开括号。例如解 2(x + 3) = x + 8。将括号展开得到 2x + 6 = x + 8，然后使用天平法合并同类项。

Remember to multiply every term inside the bracket by the number outside. In this case, both x and 3 are multiplied by 2.

记住要用括号外的数乘以括号内的每一项。在这个例子中，x 和 3 都要乘以 2。

2(x + 3) = x + 8

2x + 6 = x + 8

x = 2

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5. Unknowns on Both Sides | 未知数在等号两边

to collect the variable terms on the side with the larger coefficient. Here the coefficient of x on the left is 3, while on the right it is 1, so move the x term from the right to the left.

有些课本练习（例如编号为 p197_2 的题目）要求解 3x + 2 = x + 10 这样的方程。关键是把含变量的项集中到系数较大的一边。这里左边 x 的系数是 3，右边是 1，因此把右边的 x 项移到左边。

Step 1: Subtract x from both sides.

步骤1：两边同时减去 x。

3x + 2 − x = x + 10 − x

2x + 2 = 10

Step 2: Subtract 2 from both sides.

步骤2：两边同时减去 2。

2x = 8

Step 3: Divide both sides by 2.

步骤3：两边同时除以 2。

x = 4

Always move the smaller variable term first. This avoids negative coefficients and keeps the working simple.

总是先移系数较小的变量项。这样可以避免负系数，保持解题过程简单。

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6. Fractional Coefficients | 分数系数

e denominator first. This clears the fraction and makes the equation easier to handle, especially when the unknown appears on both sides.

如果方程含有 x/2 + 3 = x − 1 这样的分数，先让每一项都乘以分母。这样可以消去分数，使方程更容易处理，尤其是当未知数出现在等号两边时。

x/2 + 3 = x − 1

Multiply both sides by 2.

两边同时乘以 2。

x + 6 = 2x − 2

Then collect variable terms and solve.

然后合并变量项并求解。

8 = x

Always check that you have multiplied every term by the denominator, including the constant terms.

一定要检查你是否已经把每一项（包括常数项）都乘以分母。

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7. Checking Solutions by Substitution | 用代入法检验解

= 14, and the right side is 4 + 10 = 14. Both sides equal 14, so the solution is correct.

把你的答案代入原方程进行检验。对于 3x + 2 = x + 10，将 x = 4 代入：左边为 3(4) + 2 = 14，右边为 4 + 10 = 14。两边都等于 14，因此解是正确的。

Always show this checking step when a question asks you to verify your answer. It is also a useful habit for catching sign errors or arithmetic slips.

当题目要求验证答案时，要写出检验步骤。这也是发现符号错误或计算失误的好习惯。

Left: 3(4) + 2 = 12 + 2 = 14

Right: 4 + 10 = 14

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8. Word Problems and Real Contexts | 应用题与实际情境

5 cm. Its perimeter is 32 cm. Set up the equation 2(2x + 1) + 2(5) = 32, then solve for x.

应用题经常会列出线性方程。例如，一个矩形的长为 (2x + 1) cm，宽为 5 cm，周长是 32 cm。列方程 2(2x + 1) + 2(5) = 32，然后求解 x。

Always define what x stands for when you write your solution. In this case, x represents an unknown length that determines the rectangle's longer side.

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