📚 Solving Linear Equations | 解一元一次方程
Linear equations are one of the most important building blocks in KS3 mathematics. They appear in algebra, word problems, graphs and even in science calculations. A linear equation contains an unknown number, usually written as x, and the highest power of x is 1. Learning to solve these equations gives you a powerful tool for finding unknown values in real-life situations.
一元一次方程是 KS3 数学中最重要的基础内容之一。它们出现在代数、应用题、图像甚至科学计算中。一元一次方程含有一个未知数，通常用 x 表示，并且 x 的最高次数是 1。学会解这类方程能让你掌握一种强大的工具，用来解决现实生活中的未知量问题。
1. What Is a Linear Equation? | 什么是一元一次方程
A linear equation is an equation of the form ax + b = c, where a, b and c are numbers and x is the unknown variable. The word “linear” means that the variable appears only to the power of 1, so you will never see x² or x³ in this type of equation. Examples include 2x + 3 = 11, 5x − 7 = 18 and 4(x + 2) = 20.
一元一次方程是形如 ax + b = c 的方程，其中 a、b、c 是数字，x 是未知数。“一次” 表示未知数只以 1 次方的形式出现，所以在这类方程中你不会看到 x² 或 x³。例如 2x + 3 = 11、5x − 7 = 18 和 4(x + 2) = 20。
An equation is different from an expression because it contains an equals sign. The equals sign tells us that the left side and the right side have the same value. When we solve an equation, we find the value of x that makes both sides equal.
方程与表达式不同，因为它含有等号。等号告诉我们左边和右边的值相等。当我们解方程时，就是找出使两边相等的 x 的值。
2. The Balancing Method | 天平法
The key idea in solving linear equations is to keep the equation balanced. You can think of an equation as a balance scale. Whatever you do to one side, you must also do to the other side, otherwise the scale will tip. This means you can add, subtract, multiply or divide both sides by the same number without changing the solution.
解一元一次方程的关键思想是保持方程平衡。你可以把方程想象成一座天平。无论你对一边做什么，你必须对另一边做同样的操作，否则天平就会倾斜。这意味着你可以在两边同时加、减、乘或除以同一个数，而不会改变方程的解。
For example, if x + 5 = 12, we can subtract 5 from both sides to get x = 7. The operation is legal because the same number is removed from each side, so the balance is preserved.
例如，如果 x + 5 = 12，我们可以在两边同时减去 5，得到 x = 7。这个操作是合法的，因为两边都减去了同一个数，所以平衡得以保持。
3. Solving One-Step Equations | 解一步方程
One-step equations only need one operation to isolate x. For example, x + 4 = 9 is solved by subtracting 4 from both sides, giving x = 5. Similarly, x − 3 = 8 is solved by adding 3 to both sides, giving x = 11.
一步方程只需要一次运算就能把 x 单独留在一边。例如，x + 4 = 9 可以通过在两边同时减去 4 来解，得到 x = 5。类似地，x − 3 = 8 可以通过在两边同时加上 3 来解，得到 x = 11。
If the equation involves multiplication or division, use the inverse operation. For 3x = 21, divide both sides by 3 to get x = 7. For x ÷ 5 = 4, multiply both sides by 5 to get x = 20.
如果方程中含有乘法或除法，就要使用逆运算。对于 3x = 21，两边同时除以 3，得到 x = 7。对于 x ÷ 5 = 4，两边同时乘以 5，得到 x = 20。
Remember that the goal is always to leave x by itself on one side of the equation.
请记住，目标始终是让 x 单独留在方程的一边。
4. Solving Two-Step Equations | 解两步方程
Two-step equations require two operations. For example, in 2x + 3 = 11, we first subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4. The order matters: undo addition or subtraction first, then undo multiplication or division.
两步方程需要两次运算。例如，在 2x + 3 = 11 中，我们先在两边同时减去 3，得到 2x = 8，然后在两边同时除以 2，得到 x = 4。顺序很重要：先消去加法或减法，再消去乘法或除法。
Another example is 5x − 7 = 18. Add 7 to both sides: 5x = 25. Then divide both sides by 5: x = 5. Always write each step clearly so that you can check your work later.
另一个例子是 5x − 7 = 18。两边同时加上 7：5x = 25。然后两边同时除以 5：x = 5。每一步都要写清楚，这样之后可以检查你的解题过程。
5. Equations with Brackets | 含括号的方程
When an equation contains brackets, the first step is usually to expand them. For example, 3(x + 4) = 27 can be expanded to 3x + 12 = 27. Then subtract 12 from both sides to get 3x = 15, and divide by 3 to find x = 5.
当方程中含有括号时，第一步通常是展开括号。例如，3(x + 4) = 27 可以展开为 3x + 12 = 27。然后两边同时减去 12，得到 3x = 15，再除以 3，得到 x = 5。
Alternatively, you can divide both sides by the number outside the bracket first. In 3(x + 4) = 27, dividing both sides by 3 gives x + 4 = 9, so x = 5. Both methods are valid, but expanding is often safer when the numbers are not perfect multiples.
另一种方法是先在两边同时除以括号外的数字。在 3(x + 4) = 27 中，两边同时除以 3 得到 x + 4 = 9，所以 x = 5。两种方法都有效，但当数字不是完美倍数时，展开括号通常更稳妥。
6. Unknowns on Both Sides | 未知数在两边
Some equations have the unknown on both sides of the equals sign, such as 5x + 3 = 2x + 15. To solve this, first collect the x terms on one side. Subtract 2x from both sides to get 3x + 3 = 15. Then subtract 3 from both sides to get 3x = 12, and divide by 3 to get x = 4.
有些方程的未知数出现在等号两边，例如 5x + 3 = 2x + 15。解这类方程时，先把含 x 的项移到一边。两边同时减去 2x，得到 3x + 3 = 15。然后两边同时减去 3，得到 3x = 12，再除以 3，得到 x = 4。
A good habit is to move the smaller x term so that the remaining coefficient is positive. For example, in 4x − 6 = 7x + 9, subtract 4x from both sides to get −6 = 3x + 9. Then subtract 9 to get −15 = 3x, and divide by 3 to get −5 = x, or x = −5.
一个好习惯是移动较小的 x 项，使剩下的系数为正。例如，在 4x − 6 = 7x + 9 中，两边同时减去 4x，得到 −6 = 3x + 9。然后减去 9，得到 −15 = 3x，再除以 3，得到 −5 = x，即 x = −5。
7. Equations with Fractions | 含分数的方程
Equations with fractions can look difficult, but you can clear the fractions by multiplying every term by the lowest common denominator. For example, in x/3 + 1 = 5, multiply every term by 3 to get x + 3 = 15, so x = 12.
含分数的方程看起来可能很难，但你可以通过将每一项乘以最小公分母来消去分数。例如，在 x/3 + 1 = 5 中，将每一项乘以 3，得到 x + 3 = 15，所以 x = 12。
For an equation like (2x + 1)/4 = 3, multiply both sides by 4 to get 2x + 1 = 12. Then subtract 1 to get 2x = 11, and divide by 2 to get x = 5.5 or 11/2.
对于像 (2x + 1)/4 = 3 这样的方程，两边同时乘以 4，得到 2x + 1 = 12。然后减去 1，得到 2x = 11，再除以 2，得到 x = 5.5 或 11/2。
Always multiply every term, not just the fraction, otherwise the equation will become unbalanced.
一定要乘以每一项，而不仅仅是分数项，否则方程就会失去平衡。
8. Forming Equations from Word Problems | 根据应用题列方程
Many KS3 questions ask you to write an equation from a word problem. Start by letting x represent the unknown quantity. Then translate the words into mathematical operations. For example, “Three more than twice a number is 19” becomes 2x + 3 = 19, so x = 8.
许多 KS3 题目要求你根据应用题列出方程。首先设 x 代表未知量。然后把文字翻译成数学运算。例如，“一个数的两倍再加 3 等于 19” 可以写成 2x + 3 = 19，所以 x = 8。
A rectangle has length (x + 5) cm and width 4 cm. Its perimeter is 38 cm. Form the equation 2(x + 5) + 2(4) = 38. Simplify to 2x + 10 + 8 = 38, so 2x + 18 = 38, then 2x = 20, and x = 10.
一个长方形的长为 (x + 5) cm，宽为 4 cm，周长为 38 cm。列出方程 2(x + 5) + 2(4) = 38。化简得 2x + 10 + 8 = 38，所以 2x + 18 = 38，然后 2x = 20，x = 10。
The key is to identify the relationship between the given numbers and the unknown, and write that relationship using algebraic symbols.
关键是找出已知数字与未知数之间的关系，并用代数符号写出这种关系。
9. Checking Your Answer | 检验答案
After solving an equation, always substitute your answer back into the original equation to check it works. For example, if you solved 2x + 3 = 11 and got x = 4, replace x with 4: 2(4) + 3 = 8 + 3 = 11, which matches the right side.
解完方程后，一定要把你的答案代回原方程，检验它是否正确。例如，如果你解 2x + 3 = 11 得到 x = 4，就把 x 换成 4：2(4) + 3 = 8 + 3 = 11，与右边相等。
Checking is especially important in exams because it can catch sign errors or arithmetic mistakes. If the left side does not equal the right side after substitution, go back and find where the error occurred.
在考试中检验尤其重要，因为它能发现符号错误或算术错误。如果代入后左边不等于右边，就回去找出错误发生的地方。
10. Common Errors and How to Avoid Them | 常见错误及避免方法
One common mistake is forgetting to operate on both sides. For example, in x + 4 = 10, some students write x = 10 − 4 = 6 but forget to show the subtraction on both sides. This can lead to errors in more complex equations.
一个常见错误是忘记在两边同时操作。例如，在 x + 4 = 10 中，有些学生直接写 x = 10 − 4 = 6，却没有展示两边同时减去的步骤。在更复杂的方程中，这种习惯可能导致错误。
Another mistake is mishandling negative signs. In 3 − 2x = 7, subtract 3 from both sides to get −2x = 4. Dividing by −2 gives x = −2. If you divide by 2 instead, you will get the wrong sign.
另一个错误是处理负号不当。在 3 − 2x = 7 中，两边同时减去 3，得到 −2x = 4。除以 −2 得到 x = −2。如果你除以 2，就会得到错误的符号。
Finally, when expanding brackets, remember to multiply every term inside the bracket by the number outside. For 2(x + 3) = 10, expanding correctly gives 2x + 6 = 10, not 2x + 3 = 10.
最后，展开括号时，记得将括号内的每一项都乘以括号外的数字。对于 2(x + 3) = 10，正确展开是 2x + 6 = 10，而不是 2x + 3 = 10。
11. Practice Questions | 练习题
Try these questions to test your understanding. Solve each equation and check your answer by substitution.
尝试以下题目来检验你的理解。解出每个方程，并用代入法检验答案。
- 4x + 7 = 31
- 5(x − 2) = 20
- 3x − 5 = 2x + 9
- (x + 3)/4 = 6
- 7 − 2x = 1
Answers: x = 6, x = 6, x = 14, x = 21, x = 3. If you got any wrong, work through the balancing steps again.
答案：x = 6，x = 6，x = 14，x = 21，x = 3。如果有任何错误，请重新按照天平法步骤计算。
12. Summary | 总结
Solving linear equations is a core skill that will appear throughout your mathematics course. Remember to keep the equation balanced, perform inverse operations in the correct order, expand brackets carefully, and always check your solution. With regular practice, you can solve any KS3 linear equation confidently.
解一元一次方程是一项核心技能，将贯穿你的整个数学课程。记住保持方程平衡、按正确顺序进行逆运算、仔细展开括号，并始终检验你的解。通过经常练习，你就能自信地解出任何 KS3 一元一次方程。
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