📚 Solving Linear Equations | 解线性方程
Linear equations are the foundation of algebra. They allow us to find the value of an unknown number when we have an equation that can be represented by a straight line on a graph. At KS3 level, you will learn to solve equations step by step, using inverse operations to isolate the variable.
线性方程是代数的基础。当我们有一个可以用图上的直线表示的方程时，线性方程可以帮助我们求出未知数的值。在 KS3 阶段，你将学习一步步解方程，使用逆运算来分离变量。
1. What Is a Linear Equation? | 什么是线性方程？
A linear equation is an algebraic statement where the highest power of the variable is 1. It usually looks like ax + b = c or similar forms. The solution is the value of the variable that makes the equation true.
线性方程是指变量的最高次幂为 1 的代数等式。它通常形如 ax + b = c 或类似形式。解就是使方程成立的变量的值。
For example, in the equation x + 5 = 12, the solution is x = 7 because 7 + 5 = 12.
例如，在方程 x + 5 = 12 中，解是 x = 7，因为 7 + 5 = 12。
Linear equations can have one solution, no solution, or infinitely many solutions. In KS3, we focus on equations with one unique solution.
线性方程可能有一个解、无解或无穷多个解。在 KS3，我们主要关注有唯一解的方程。
2. Using Inverse Operations | 使用逆运算
To solve an equation, we undo the operations that are applied to the variable. Addition and subtraction are inverse operations, as are multiplication and division. We must always keep the equation balanced by doing the same thing to both sides.
要解方程，我们要逆过来做对变量进行的运算。加法和减法互为逆运算，乘法和除法也互为逆运算。我们必须始终保持方程平衡，在等号两边做相同的操作。
Consider 3x = 15. The inverse of multiplying by 3 is dividing by 3. So we divide both sides by 3: x = 5.
考虑 3x = 15。乘以 3 的逆运算是除以 3。所以两边同时除以 3：x = 5。
For x - 4 = 9, we add 4 to both sides, getting x = 13. Always check your answer by substituting it back into the original equation.
对于 x - 4 = 9，两边同时加 4，得到 x = 13。一定要把答案代回原方程检验。
Inverse operations are the key to isolating the variable on one side of the equation.
逆运算是将变量分离到方程一边的关键。
3. Solving Simple One-Step Equations | 解简单的一步方程
A one-step equation requires only one inverse operation to find the solution. For example, x + 7 = 10 is solved by subtracting 7 from both sides, giving x = 3.
一步方程只需要一次逆运算就能求出解。例如，x + 7 = 10 的解是两边同时减去 7，得到 x = 3。
Another type: 5x = 35. Divide both sides by 5: x = 7.
另一种类型：5x = 35。两边同时除以 5：x = 7。
Equations with division, like x/4 = 3, are solved by multiplying both sides by 4, yielding x = 12.
带除法的方程，如 x/4 = 3，两边同时乘以 4，得到 x = 12。
These equations build confidence in maintaining balance across the equals sign.
这些方程帮助建立保持等号两边平衡的信心。
4. Solving Two-Step Equations | 解两步方程
Two-step equations involve two operations. For instance, 2x + 3 = 11. First, subtract 3 from both sides to get 2x = 8. Then divide both sides by 2, giving x = 4.
两步方程包含两种运算。例如，2x + 3 = 11。首先，两边同时减去 3，得到 2x = 8。然后两边同时除以 2，得出 x = 4。
Always undo the addition or subtraction first, then multiplication or division. This follows the reverse order of operations (BIDMAS/BODMAS).
总是先逆过来做加法或减法，再做乘法或除法。这遵循运算顺序的逆向（BIDMAS/BODMAS）。
Example: y/5 - 2 = 4. Add 2 to both sides: y/5 = 6. Multiply both sides by 5: y = 30.
例子：y/5 - 2 = 4。两边加 2：y/5 = 6。两边乘以 5：y = 30。
The key is to isolate the term containing the variable before dealing with the coefficient.
关键是在处理系数之前先分离出包含变量的项。
5. Equations with Brackets | 带括号的方程
When an equation contains brackets, expand them first using the distributive law or simplify by dividing if there's a common factor. For example, 3(x + 2) = 15. You can expand to 3x + 6 = 15, then solve.
当方程含有括号时，先用分配律展开，或者如果有公因子可以两边同时除以该因子来简化。例如，3(x + 2) = 15。你可以展开成 3x + 6 = 15，然后求解。
Alternatively, divide both sides by 3 first: x + 2 = 5, then x = 3. Both methods give the same answer.
或者，先两边同时除以 3：x + 2 = 5，然后 x = 3。两种方法得出相同答案。
For 2(3y - 4) = 10, expand: 6y - 8 = 10. Add 8 to get 6y = 18, so y = 3.
对于 2(3y - 4) = 10，展开：6y - 8 = 10。加 8 得 6y = 18，所以 y = 3。
Remember, a minus sign before a bracket changes the signs of all terms inside when expanding.
记住，括号前的减号展开时会改变括号内所有项的符号。
6. Variables on Both Sides | 变量在等号两边
Equations like 5x + 2 = 3x + 10 have the unknown on both sides. To solve, first collect all variable terms on one side by adding or subtracting. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8, so x = 4.
像 5x + 2 = 3x + 10 这样的方程，未知数在等号两边。要解这类方程，首先通过加减将所有含变量的项移到一边。两边同时减去 3x：2x + 2 = 10。然后减 2：2x = 8，所以 x = 4。
You could also subtract 5x: 2 = -2x + 10, leading to -2x = -8, and dividing by -2 gives x = 4. Choose the method that avoids negative coefficients if possible.
你也可以减去 5x：2 = -2x + 10，得出 -2x = -8，除以 -2 得 x = 4。尽量选择避免负系数的方法。
Always aim to have a positive variable term in the final step for simplicity.
为简单起见，最后一步尽量让变量项为正。
7. Equations with Fractions | 含有分数的方程
When equations involve fractions, it often helps to multiply every term by the lowest common denominator (LCD) to clear the fractions. Example: x/3 + 2 = 5. Multiply all terms by 3: x + 6 = 15, so x = 9.
当方程含有分数时，通常帮助把每一项乘以最小公分母来清除分数。例子：x/3 + 2 = 5。所有项乘以 3：x + 6 = 15，所以 x = 9。
For (2x)/5 = 4, multiply both sides by 5: 2x = 20, then x = 10.
对于 (2x)/5 = 4，两边乘以 5：2x = 20，然后 x = 10。
If the equation has multiple fractions, find the LCD of all denominators and multiply every term by it. This transforms the equation into a simpler one without fractions.
如果方程有多个分数，找出所有分母的最小公分母，然后把每一项都乘以它。这会将方程转化为没有分数的更简单形式。
8. Checking Your Solution | 检验你的解
After finding a solution, substitute it back into the original equation to verify. For 4x - 7 = 13, if you got x = 5 5) - 7 = 20 - 7 = 13, which matches the right side. This confirms the solution is correct.
找到解之后，把它代回原方程验证。对于 4x - 7 = 13，如果你得到 x = 5，检验：4(5) - 7 = 20 - 7 = 13，与右边相等。这确认了答案正确。
Checking is especially important when dealing with negative numbers or fractions to ensure no arithmetic mistake.
在处理负数或分数时检验尤其重要，以确保没有计算错误。
This habit will serve you well as equations become more complex in higher years.
这个习惯在以后更高年级碰到更复杂方程时会很有帮助。
9. Forming Equations from Word Problems | 根据应用题建立方程
Word problems require translating a real-life scenario into a linear equation. Identify the unknown quantity and assign it a variable (often x). Then, write an equation based on the given relationships.
应用题要求将现实情景转化为线性方程。找出未知量，并用一个变量（通常是 x）表示。然后，根据给定的关系写出方程。
Example: "Three more than twice a number is 11." Let the number be x. Twice the number is 2x. Three more than that is 2x + 3. The equation: 2x + 3 = 11, solving gives x = 4.
例子："一个数的两倍再加 3 等于 11。" 设这个数为 x。两倍就是 2x。再加 3 就是 2x + 3。方程：2x + 3 = 11，解得 x = 4。
Practice translating phrases: "less than" means subtraction, "more than" means addition, "times" means multiplication, "divided by" means division.
练习翻译短语："小于"是减法，"大于"是加法，"倍"是乘法，"除以"是除法。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
A typical error is forgetting to perform the same operation on both sides, which unbalances the equation. Always remember the golden rule: whatever you do to one side, you must do to the other.
一个典型错误是忘记在等号两边进行相同运算，这会使方程失衡。永远记住黄金法则：对一边做什么，另一边也必须做同样的事。
Another mistake involves signs when expanding brackets with a minus, e.g., -(2x - 3) = -2x + 3, not -2x - 3. Take care with negative numbers.
另一个错误是展开带减号的括号时的符号，例如 -(2x - 3) = -2x + 3，而不是 -2x - 3。注意负数的处理。
When moving terms, students sometimes move a term but forget to change its sign. Remember, moving a term to the other side changes its sign.
在移项时，学生有时移了项但忘记变号。记住，把一项移到另一边要改变它的符号。
Always double-check your arithmetic, especially when dealing with fractions and decimals.
永远要仔细检查计算，尤其是在处理分数和小数时。
11. Summary and Key Takeaways | 总结与关键要点
Linear equations are solved by isolating the variable using inverse operations and maintaining balance. You learned to handle one-step, two-step, bracket, variable-on-both-sides, and fraction equations. Always expand brackets first, collect like terms, then isolate the variable step by step.
解线性方程是通过使用逆运算和保持平衡来分离变量。你学会了处理一步、两步、带括号、变量在两边和带分数的方程。总是先展开括号，合并同类项，然后一步步隔离变量。
Key skills: using inverse operations, balancing equations, expanding brackets correctly, and checking solutions by substitution. These are essential for all future algebra topics.
关键技能：使用逆运算，平衡方程，正确展开括号，并通过代入检验解。这些都是未来所有代数主题的基础。
With practice, solving linear equations will become automatic, giving you confidence to tackle more advanced mathematics.
通过练习，解线性方程将变得自动，让你有信心挑战更高级的数学。
Published by TutorHao | Cambridge KS3 Mathematics Revision Series | aleveler.com
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