Solving Linear Equations | 解线性方程

📚 Solving Linear Equations | 解线性方程

Linear equations are the foundation of algebra. In this article, we will explore how to solve them step by step, from simple one-step equations to equations involving brackets, fractions and variables on both sides. You will also learn how to apply these skills to real-life word problems.

线性方程是代数的基础。在这篇文章中，我们将一步步探索如何解线性方程，从简单的一步方程到含有括号、分数以及两边都有变量的方程。你还将学习如何将这些技能应用到实际生活应用题中。

1. What Is a Linear Equation? | 什么是线性方程？

A linear equation is an equation where the highest power of the variable is 1. For example, 3x + 5 = 14 is linear because x appears only to the power of 1. If x² appears, the equation is quadratic, not linear.

线性方程是变量最高次数为1的方程。例如，3x + 5 = 14 是线性的，因为 x 只以一次方的形式出现。如果出现 x²，那它就是二次方程，而不是线性方程。

The general form of a linear equation in one variable is ax + b = c, where a, b and c are constants and a ≠ 0.
一元线性方程的一般形式是 ax + b = c，其中 a、b、c 是常数，且 a ≠ 0。
The solution of a linear equation is the value of the variable that makes the equation true.
线性方程的解是使方程成立的变量的值。

2x + 3 = 11 ⇒ x = 4

2. The Golden Rule: Keep It Balanced | 黄金法则：保持平衡

An equation is like a balance scale. Whatever you do to one side, you must do to the other side. This keeps the two sides equal.

方程就像一架天平。你对一边做了什么，就必须对另一边做同样的事。这样才能保持两边相等。

For example, to solve x + 5 = 12, subtract 5 from both sides:

例如，解 x + 5 = 12 时，两边同时减去5：

x + 5 − 5 = 12 − 5 ⇒ x = 7

This operation is called inverse operations: addition and subtraction are inverses, multiplication and division are inverses.

这种运算称为逆运算：加法与减法互为逆运算，乘法与除法互为逆运算。

3. Solving One-Step Equations | 解一步方程

One-step equations require only one operation to solve. Let us look at four types:

一步方程只需要一步运算即可求解。我们来看四种类型：

Addition/Subtraction: x − 4 = 10 ⇒ x = 14.
加减法： x − 4 = 10 ⇒ x = 14。
Multiplication: 5x = 35 ⇒ x = 7.
乘法： 5x = 35 ⇒ x = 7。
Division: x/3 = 6 ⇒ x = 18.
除法： x/3 = 6 ⇒ x = 18。
Fractional coefficient: (2/3)x = 8 ⇒ x = 12.
分数系数： (2/3)x = 8 ⇒ x = 12。

Always check your answer by substituting it back into the original equation.

务必把答案代回原方程进行检验。

4. Equations with Variables on Both Sides | 两边都有变量的方程

When the variable appears on both sides, collect the variable terms on one side and constant terms on the other. Usually, we put the variable on the left side.

当变量出现在两边时，把含变量的项集中到一边，常数项集中到另一边。通常我们把变量放在左边。

Solve 5x − 2 = 3x + 8:

解 5x − 2 = 3x + 8：

5x − 3x = 8 + 2 ⇒ 2x = 10 ⇒ x = 5

Subtract 3x from both sides to eliminate x from the right.
两边同时减去 3x，消去右边的 x。
Add 2 to both sides to remove the constant on the left.
两边同时加上2，消去左边的常数。
Divide by the coefficient of x.
除以 x 的系数。

Check: 5(5) − 2 = 23 and 3(5) + 8 = 23 ✓

检验：5(5) − 2 = 23，3(5) + 8 = 23 ✓

5. Equations with Brackets | 含有括号的方程

To solve equations with brackets, first expand the brackets using the distributive property, then simplify and solve.

解含有括号的方程时，先用乘法分配律展开括号，再化简求解。

Solve 3(2x − 1) = 15:

解 3(2x − 1) = 15：

6x − 3 = 15 ⇒ 6x = 18 ⇒ x = 3

If there are brackets on both sides, expand them first and then collect like terms.

如果两边都有括号，先展开，再合并同类项。

Solve 2(x + 3) = 4(x − 1):

解 2(x + 3) = 4(x − 1)：

2x + 6 = 4x − 4 ⇒ 10 = 2x ⇒ x = 5

6. Equations with Fractions | 含有分数的方程

Fractions can be eliminated by multiplying both sides of the equation by the lowest common multiple (LCM) of all denominators. This process is called clearing fractions.

分数可以通过将方程两边同时乘以所有分母的最小公倍数（LCM）来消除。这一过程称为“去分母”。

Solve x/3 + 1 = x/2:

解 x/3 + 1 = x/2：

Multiply both sides by 6 (the LCM of 3 and 2):

两边同时乘以6（3和2的最小公倍数）：

6 × (x/3 + 1) = 6 × (x/2) ⇒ 2x + 6 = 3x ⇒ x = 6

Be careful to multiply every term by the LCM, not just the fractions.
注意：要把每一项都乘以LCM，而不仅仅是分数项。
If the equation contains terms like (x+2)/4, multiply the whole numerator by the LCM.
如果方程中含有像 (x+2)/4 这样的项，要将整个分子乘以LCM。

7. Changing the Subject of a Formula | 改变公式的主项

Rearranging a formula is exactly like solving an equation. You perform the same inverse operations until the required variable stands alone on one side.

重排公式和解方程完全一样。你执行相同的逆运算，直到要求的变量单独出现在一边。

Make x the subject of y = 5x + 2:

将 y = 5x + 2 化为 x 的表达式：

y − 2 = 5x ⇒ x = (y − 2) / 5

For formula involving fractions, clear denominators first. Make r the subject of C = 2πr:

对于含分数的公式，先去分母。将 C = 2πr 化为 r 的表达式：

C = 2πr ⇒ r = C / (2π)

Formula	Subject to find	Result
A = lw	l	l = A/w
v = u + at	a	a = (v − u)/t
P = 2(a + b)	b	b = P/2 − a

8. Solving Word Problems | 解应用题

Real-life problems can often be translated into linear equations. Follow these steps:

现实生活问题通常可以转化为线性方程。遵循以下步骤：

Read the problem carefully and identify the unknown quantity. Assign it a variable, e.g., x.
仔细阅读题目并找出未知量。为其设一个变量，例如 x。
Translate the conditions of the problem into an equation.
将题目中的条件转化为方程。
Solve the equation using the methods above.
用上述方法解方程。
Check whether your answer makes sense in the context of the problem.
检查答案在题目背景下是否合理。

Example: The sum of three consecutive integers is 36. Find the integers.

例子：三个连续整数的和是36，求这三个整数。

Let the first integer be x. Then the next two are x + 1 and x + 2.

设第一个整数为 x，那么接下来两个是 x + 1 和 x + 2。

x + (x + 1) + (x + 2) = 36 ⇒ 3x + 3 = 36 ⇒ 3x = 33 ⇒ x = 11

The integers are 11, 12 and 13.

这三个整数是 11、12 和 13。

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

Many students make similar mistakes when solving linear equations. Here are the most common ones:

许多学生在解线性方程时会犯类似的错误。以下是最常见的几种：

Mistake 1: Forgetting to perform the same operation on both sides.
错误1：忘记在两边进行相同的运算。
Mistake 2: Incorrect sign when moving terms across the equals sign.
错误2：移项时符号错误。
Mistake 3: Not expanding brackets correctly.
错误3：展开括号不正确。
Mistake 4: Multiplying only the fraction terms when clearing denominators.
错误4：去分母时只乘以分数项。
Mistake 5: Forgetting to check the solution.
错误5：忘记检验解。

Always write down each step clearly and check your final answer by substitution.

始终清晰地写出每一步，并通过代入法检查最终答案。

10. Practice Questions | 练习题目

Try these questions to test your understanding. Solve each equation and then check your answers below.

尝试以下题目以检验你的理解。解每个方程，然后在下方对答案。

7x − 4 = 17
4(x + 2) = 28
5x + 3 = 2x + 15
x/2 + 3 = 8
2(3x − 5) = 7x + 1

Question	Answer
1	x = 3
2	x = 5
3	x = 4
4	x = 10
5	x = −11

If you solved them correctly, well done! If not, revisit the relevant section.

如果你全部正确，恭喜你！如果有错，请回顾相应的小节。

Solving linear equations is a vital skill for IGCSE Mathematics. Once you master the balance method, brackets and fractions, you will be able to tackle a wide range of algebra problems with confidence.

解线性方程是 IGCSE 数学中一项至关重要的技能。一旦你掌握了平衡法、括号和分数的处理，你就能自信地应对各种代数问题。

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Solving Linear Equations | 解线性方程