Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Solving linear equations is a core IGCSE algebra skill. It allows you to find the unknown value that makes an equation true, using inverse operations and the balance method.

解一元一次方程是 IGCSE 代数中的核心技能。它让你通过逆运算和天平法求出使等式成立的未知数值。

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

A linear equation has variables raised only to the power of 1. Its graph is a straight line.

一元一次方程中，未知数的最高次数仅为 1。它的图像是一条直线。

General form of a linear equation in one variable:

一元一次方程的一般形式为：

ax + b = 0, a ≠ 0

Examples include 2x + 6 = 14, 5y − 3 = 2y + 9, and x/4 + 1 = 3.

例如：2x + 6 = 14、5y − 3 = 2y + 9、x/4 + 1 = 3。

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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 also do to the other side.

把等式看作一架平衡的天平。你对一边做任何运算，另一边也必须做同样的运算。

• Add or subtract the same number from both sides.

  两边同时加上或减去同一个数。

• Multiply or divide both sides by the same non-zero number.

  两边同时乘以或除以同一个不为零的数。

This keeps the equation equivalent to the original equation, so the solution does not change.

这样方程仍与原方程同解，因此解不会改变。

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3. One-Step Equations | 一步方程

Use one inverse operation to isolate the variable.

使用一个逆运算就可以把未知数单独留在一边。

Example: x + 7 = 15.

例子：x + 7 = 15。

x + 7 − 7 = 15 − 7 → x = 8

Example: 4x = 28.

例子：4x = 28。

4x ÷ 4 = 28 ÷ 4 → x = 7

Operation in equation	Inverse operation to use
+ a	− a
− a	+ a
× a	÷ a
÷ a	× a

The inverse operation reverses the one that is applied to the variable.

逆运算可以抵消作用在未知数上的运算。

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4. Two-Step Equations | 两步方程

For equations like 3x + 5 = 20, undo the addition or subtraction first, then undo multiplication or division.

对于 3x + 5 = 20 这类方程，先消去加或减，再消去乘或除。

Worked solution:

解题过程：

3x + 5 − 5 = 20 − 5 → 3x = 15 → x = 5

Always reverse the order of operations: start with the outside operation.

一定要逆着运算顺序：先处理最外层的运算。

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5. Expanding Brackets First | 先去括号

If an equation contains brackets, expand them before using the balance method.

如果方程中含有括号，先用乘法分配律去括号，再使用天平法。

Example: 2(x + 3) = 16.

例子：2(x + 3) = 16。

2x + 6 = 16 → 2x = 10 → x = 5

Be careful when a negative sign is in front of a bracket: −(x − 4) becomes −x + 4.

括号前有负号时要特别小心：−(x − 4) 应变为 −x + 4。

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6. Variables on Both Sides | 未知数在两边

When variables appear on both sides, collect all variable terms on one side and constants on the other.

当未知数出现在两边时，把所有的未知数项移到一边，常数项移到另一边。

Example: 7x − 4 = 3x + 8.

例子：7x − 4 = 3x + 8。

7x − 3x = 8 + 4 → 4x = 12 → x = 3

Remember: when you move a term across the equals sign, its sign changes.

记住：把一项移到等号另一边时，符号要改变。

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7. Equations with Fractions | 含分数的方程

Multiply every term by the lowest common denominator to clear fractions.

用最小公分母乘以每一项，以消去分母中的分数。

Example: x/3 + 2 = 7.

例子：x/3 + 2 = 7。

3 × (x/3 + 2) = 3 × 7 → x + 6 = 21 → x = 15

Example: (2x + 1)/5 = 3.

例子：(2x + 1)/5 = 3。

2x + 1 = 15 → 2x = 14 → x = 7

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8. Checking Your Answer | 检验答案

Substitute your solution back into the original equation to confirm both sides are equal.

把解代入原方程，确认两边相等。

Example: check x = 5 in 3x + 5 = 20.

例子：在 3x + 5 = 20 中检验 x = 5。

3 × 5 + 5 = 15 + 5 = 20 ✓

Checking is especially important in exams to avoid sign or arithmetic errors.

考试中检验尤其重要，可以避免符号或计算错误。

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9. Linear Equations in Word Problems | 应用题中的一次方程

Translate the words into an equation, solve it, and interpret the answer in context.

把文字转化为方程，求解，并结合题意解释答案。

Example: The sum of three consecutive numbers is 42. Find the numbers.

例子：三个连续数的和是 42。求这三个数。

Let the numbers be x, x + 1, and x + 2.

设三个数为 x、x + 1、x + 2。

x + (x + 1) + (x + 2) = 42 → 3x + 3 = 42 → x = 13

The numbers are 13, 14, and 15.

三个数是 13、14、15。

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10. Common Mistakes to Avoid | 常见错误

• Forgetting to apply an operation to both sides.

  忘记对两边同时进行同一运算。

• Changing the sign incorrectly when moving terms.

  移项时符号改变错误。

• Expanding brackets with negative signs carelessly.

  括号前为负号时去括号粗心。

• Multiplying only some terms by the common denominator.

  乘以公分母时只乘了部分项。

Slow down, show working, and check your answer to avoid these errors.

减慢速度、写出步骤，并检验答案，以避免这些错误。

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11. Summary and Exam Tips | 总结与考试提示

Use the balance method, clear brackets and fractions first, collect variable terms on one side, then check.

使用天平法，先去括号和分母，把未知数项移到一边，然后检验。

• Show every step to earn method marks.

  写出每一步以取得过程分。

• Write the final answer clearly as x = …

  最后答案清楚地写成 x = …

• Substitute back to check your solution.

  把解代回原式进行检验。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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