Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the foundation of algebra, appearing whenever we find an unknown number that makes a mathematical statement true. In KS3 Mathematics, solving equations like 2x + 5 = 13 begins with understanding balance, inverse operations, and the idea that what we do to one side we must also do to the other.

线性方程是代数的基础，每当我们寻找一个使数学等式成立的未知数时，就会用到它。在 KS3 阶段的数学课程里，解 2x + 5 = 13 这样的方程，需要先理解天平的平衡思想、逆运算，以及“对等式一边做什么，另一边也要做同样操作”的核心原则。

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

A linear equation is an algebraic statement in which the unknown appears only to the power of 1. It can be written in the form ax + b = c, where a, b and c are known numbers and x is the variable we need to find.

一元一次方程是一种代数陈述，其中未知数的指数只出现一次方。它可以写成 ax + b = c 的形式，其中 a、b 和 c 是已知数，x 是需要我们求出的变量。

Solving the equation means finding the value of x that makes both sides equal. There is exactly one solution for a standard linear equation with one variable.

解方程意味着找出那个让等号两边相等的 x 的值。一个标准的一元一次方程只有一个解。

2. The Balance Method | 天平法

Think of an equation as a pair of balanced scales. Both sides have the same total weight. To keep the scales balanced, whatever operation you perform on one side must also be performed on the other.

把方程想象成一台平衡的天平，两边总重量相等。为了保持天平平衡，你对一边进行的任何操作，另一边也必须完全一样地进行。

For example, with x + 3 = 10, removing 3 from the left means you must also remove 3 from the right, giving x = 7.

例如，对于 x + 3 = 10，从左边拿掉 3，就必须也从右边拿掉 3，得到 x = 7。

3. Inverse Operations | 逆运算

To isolate the unknown, we use inverse operations: addition and subtraction undo each other, while multiplication and division undo each other.

为了把未知数单独留在等式一边，我们使用逆运算：加法和减法互为逆运算，乘法和除法也互为逆运算。

In 2x = 14, the x is multiplied by 2, so we divide both sides by 2: x = 14 ÷ 2 = 7.

在 2x = 14 中，x 被乘了 2，因此我们把两边同时除以 2：x = 14 ÷ 2 = 7。

4. Solving One-Step Equations | 解一步方程

Equations such as x + 8 = 15 or 3x = 21 require only one inverse operation. Subtract 8 or divide by 3, and the solution appears immediately.

像 x + 8 = 15 或 3x = 21 这样的方程只需要一步逆运算。减去 8 或除以 3，答案马上就出来了。

Always check your solution by substituting it back into the original equation. If 7 + 8 = 15, then x = 7 is correct.

一定要把得到的解代回原方程去检验。如果 7 + 8 = 15，那么 x = 7 就是正确的。

5. Solving Two-Step Equations | 解两步方程

A two-step equation like 2x + 5 = 13 involves two operations. The order of undoing follows the reverse of BIDMAS: first deal with addition or subtraction, then multiplication or division.

像 2x + 5 = 13 这样的两步方程包含两个运算。逆向求解的顺序与运算优先级顺序相反：先处理加减，再处理乘除。

Subtract 5 from both sides: 2x = 8. Then divide by 2: x = 4.

两边先减去 5：2x = 8，然后除以 2：x = 4。

Sometimes the unknown value is negative. For example, x + 7 = 2 leads to x = 2 − 7 = −5.

有时未知数的值是负数。例如 x + 7 = 2 会得出 x = 2 − 7 = −5。

Negative solutions are perfectly valid and appear often when a larger number is subtracted from a smaller one.

负数是完全合理的解，当从较小的数中减去较大的数时，经常出现这种情况。

7. Equations with the Unknown on Both Sides | 未知数在等式两边的方程

When x appears on both sides, like 5x − 3 = 2x + 9, the aim is to collect all x terms on one side and all numbers on the other.

当 x 出现在等式两边时，比如 5x − 3 = 2x + 9，目标是把所有含 x 的项移到一边，所有数字移到另一边。

Subtract 2x from both sides: 3x − 3 = 9. Then add 3: 3x = 12, so x = 4.

两边减去 2x：3x − 3 = 9，再加 3：3x = 12，因此 x = 4。

8. Expanding Brackets First | 先展开括号

If an equation contains brackets, such as 3(x + 2) = 21, you must expand the brackets before solving. Multiply the term outside by each term inside.

如果方程中含有括号，如 3(x + 2) = 21，必须先展开括号再求解。把外面的项乘以括号内的每一项。

3x + 6 = 21, then 3x = 15, so x = 5.

3x + 6 = 21，然后 3x = 15，所以 x = 5。

9. Fractions in Equations | 方程中的分数

When an equation includes fractions, multiply every term on both sides by the lowest common multiple of the denominators. This clears the fractions and makes the equation easier to solve.

当方程中含有分数时，把等式两边每一项都乘以分母的最小公倍数。这样可以消去分母，使方程更容易求解。

For x/3 + 2 = 5, multiply all terms by 3: x + 6 = 15, so x = 9.

对于 x/3 + 2 = 5，所有项乘以 3：x + 6 = 15，因此 x = 9。

10. Forming Equations from Word Problems | 根据文字题建立方程

Real-life problems often require you to write your own equation. Identify the unknown, give it a letter, and translate the words into mathematical operations.

现实生活中的问题往往需要你自己写出方程。先确定未知数，用字母表示，然后把文字转换成数学运算。

“I think of a number, double it and add 7. The result is 25.” Let the number be n: 2n + 7 = 25, so n = 9.

“我想一个数，把它加倍再加 7，结果是 25。”设这个数为 n：2n + 7 = 25，所以 n = 9。

11. Common Mistakes and How to Avoid Them | 常见错误及如何避免

A frequent error is forgetting to perform the same operation on both sides. Another is not reversing the sign correctly when moving terms across the equals sign.

一个常见错误是忘记在等式两边执行相同的操作。另一个错误是在把项移到等号另一边时没有正确地变号。

Write each step clearly and check your answer by substitution. Keeping work neat reduces mistakes.

每一步都要写清楚，并通过代入原方程检验答案。保持书写整洁可以减少错误。

12. Practice and Mastery | 练习与精通

The key to mastering linear equations is regular practice with a variety of types: one-step, two-step, with brackets, fractions, and unknowns on both sides. Build confidence by starting simple and increasing difficulty gradually.

掌握一元一次方程的关键在于对各种类型进行有规律的练习：一步方程、两步方程、含括号的、含分数的以及未知数在等式两边的。从简单开始，逐步增加难度，建立信心。

With a solid grasp of the balance method and inverse operations, you will be ready for more advanced algebra topics in KS3 and beyond.

扎实掌握天平法和逆运算后，你就为 KS3 及以后的更高阶代数内容做好了准备。

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Solving Linear Equations | 解一元一次方程