Solving Linear Equations and Applications | 线性方程的解法与应用

📚 Solving Linear Equations and Applications | 线性方程的解法与应用

Linear equations form the foundation of algebra and appear in almost every area of A-level mathematics, from coordinate geometry to calculus. Mastering their solution is essential for exam success.

线性方程是代数的基础，在A-level数学的几乎每一个领域——从坐标几何到微积分——都会出现。掌握线性方程的解法对考试成功至关重要。

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

A linear equation is an equation where the variable (usually x) appears only to the first power. The standard form is:

线性方程是指变量（通常为 x）仅以一次幂出现的方程。其标准形式为：

ax + b = 0, where a ≠ 0

Here, a and b are constants. Examples include x + 3 = 7, 2x − 5 = 11, and 4(x − 1) = 2x + 6. The goal is always to isolate the variable.

其中 a 和 b 为常数。例如 x + 3 = 7、2x − 5 = 11 和 4(x − 1) = 2x + 6。解方程的目标始终是将变量单独隔离出来。

It is important to recognise that the word ‘linear’ comes from the fact that the graph of y = ax + b is a straight line.

值得注意的是，”线性”一词来源于函数 y = ax + b 的图像是一条直线这一事实。

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2. The Balance Method | 平衡法

The balance method is the most important technique for solving linear equations. It is based on the idea that an equation is like a set of weighing scales: both sides must remain equal at all times. Therefore, any operation performed on one side must also be performed on the other.

平衡法是解线性方程最重要的技巧。它基于这样一个理念：方程就像一架天平，两边必须始终保持相等。因此，对一边执行的任何操作，也必须对另一边执行。

The operations we can use include adding, subtracting, multiplying, and dividing by the same nonzero number on both sides. For instance, if x − 4 = 9, add 4 to both sides to get x = 13.

我们可以使用的操作包括两边同时加、减、乘、除以同一个非零数。例如，若 x − 4 = 9，两边同时加4得到 x = 13。

This method is systematic and reliable, and it forms the basis for all subsequent equation-solving techniques.

这种方法系统且可靠，是后续所有解方程技巧的基础。

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3. Solving One-Step Equations | 解一步方程

One-step equations require only a single operation to solve. Consider the equation x + 5 = 12. Since 5 is added to x, we subtract 5 from both sides:

一步方程只需一次运算即可求解。考虑方程 x + 5 = 12。由于 x 加上了5，我们两边同时减去5：

x + 5 − 5 = 12 − 5 → x = 7

Similarly, if the equation is 3x = 18, we divide both sides by 3:

类似地，若方程为 3x = 18，我们两边同时除以3：

3x ÷ 3 = 18 ÷ 3 → x = 6

For subtraction and division cases, the same principle applies: use the inverse operation. If x − 7 = 10, add 7; if x ÷ 4 = 5, multiply by 4.

对于减法和除法的情况，同样的原则适用：使用逆运算。若 x − 7 = 10，两边加7；若 x ÷ 4 = 5，两边乘以4。

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4. Solving Two-Step Equations | 解两步方程

Two-step equations involve two operations and must be solved in the correct order. Consider 2x + 3 = 11. We must first remove the constant term by subtracting 3 from both sides, and then divide by the coefficient of x:

两步方程包含两个运算，必须按照正确的顺序求解。考虑 2x + 3 = 11。我们必须先通过两边同时减3来去掉常数项，然后除以 x 的系数：

2x + 3 − 3 = 11 − 3 → 2x = 8 → x = 4

The general rule is: deal with addition and subtraction before multiplication and division. This is the reverse of the order of operations (BIDMAS) used for simplifying expressions.

一般规则是：先处理加减法，再处理乘除法。这与化简表达式时使用的运算顺序（BIDMAS）正好相反。

Consider another example: x/3 − 2 = 4. First add 2 to both sides to obtain x/3 = 6, then multiply both sides by 3 to obtain x = 18.

再看一个例子：x/3 − 2 = 4。首先两边加2得到 x/3 = 6，然后两边乘以3得到 x = 18。

Always check your answer by substituting it back into the original equation. For x = 4 in the first example: 2(4) + 3 = 11, which is correct.

始终通过将答案代回原方程来检查。在第一个例子中，x = 4：2(4) + 3 = 11，正确无误。

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5. Variables on Both Sides | 变量出现在方程两边

When the variable appears on both sides of the equation, we must first collect all variable terms on one side and all constant terms on the other. Consider:

当变量出现在方程两边时，我们必须先将所有含变量的项移到一边，所有常数项移到另一边。考虑：

5x − 4 = 2x + 8

Subtract 2x from both sides: 3x − 4 = 8. Then add 4 to both sides: 3x = 12. Finally divide by 3: x = 4.

两边同时减去2x：3x − 4 = 8。然后两边加4：3x = 12。最后除以3：x = 4。

A useful strategy is to always move the smaller variable term so that the coefficient of x remains positive. This reduces the chance of sign errors.

一个有用的策略是始终移动较小的变量项，使 x 的系数保持为正。这可以减少符号错误的机会。

For equations with the variable on both sides and brackets, such as 3(x + 2) = 2(x + 5), first expand the brackets: 3x + 6 = 2x + 10. Then subtract 2x and subtract 6: x = 4.

对于两边都有变量且含括号的方程，例如 3(x + 2) = 2(x + 5)，首先展开括号：3x + 6 = 2x + 10。然后两边同时减去2x和6：x = 4。

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6. Equations with Fractions | 含分数的方程

Fractions can make equations look more complicated than they are. The best approach is to eliminate the denominators by multiplying both sides of the equation by the lowest common multiple (LCM) of all denominators.

分数会使方程看起来比实际更复杂。最好的方法是两边同时乘以所有分母的最小公倍数（LCM）来消去分母。

Consider the equation:

考虑方程：

x/2 + x/3 = 5

The LCM of 2 and 3 is 6. Multiply both sides by 6:

2和3的最小公倍数是6。两边同时乘以6：

6 × (x/2) + 6 × (x/3) = 6 × 5 → 3x + 2x = 30 → 5x = 30 → x = 6

When dealing with decimals, a similar technique applies. For instance, 0.5x + 0.25 = 1.5 can be multiplied through by 100 to obtain 50x + 25 = 150, then solved as usual.

处理小数时，也适用类似的技巧。例如，0.5x + 0.25 = 1.5 可以两边同时乘以100得到 50x + 25 = 150，然后按常规方法求解。

Fractions with denominators that are expressions, such as 2/(x + 1) = 3, can also be handled by cross-multiplication: 2 = 3(x + 1), then x = −1/3.

分母为表达式的分数方程，例如 2/(x + 1) = 3，也可以通过交叉相乘来处理：2 = 3(x + 1)，

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