📚 Solving Equations with Brackets | 解带括号的方程
Solving equations is a fundamental skill in algebra, and brackets often appear to group terms together. This guide will walk you through the steps needed to solve linear equations involving brackets, using the distributive property and careful balancing. By the end, you will be confident in expanding brackets and isolating the unknown variable to find the correct solution.
解方程是代数中的基本技能，而括号经常被用来将项组合在一起。本指南将带你逐步掌握解含有括号的线性方程所需的步骤，包括运用分配律和仔细进行等式平衡。学完本节后，你将能够自信地展开括号并分离未知变量，从而找到正确的解。
1. What is an Equation? | 什么是方程？
tains an unknown value, usually represented by a letter like x or y. The goal when solving an equation is to find the value of the unknown that makes the statement true.
方程是一个数学陈述，用等号（=）表示两个表达式相等。它包含一个未知值，通常用字母（如 x 或 y）表示。解方程的目标就是找到使该陈述成立的未知数的值。
15, the bracket indicates that the entire quantity x + 2 is multiplied by 3. To solve it, we need to remove the bracket correctly and then isolate x.
例如，在方程 3(x + 2) = 15 中，括号表示整个 x + 2 都被乘以 3。要解这个方程，我们需要正确地去掉括号，然后分离出 x。
2. Understanding Brackets | 理解括号
Brackets (parentheses) are used in algebra to group terms together. They tell us that the operation outside the bracket applies to every term inside. This is a key idea because if you ignore the brackets or handle them incorrectly, your solution will be wrong.
括号（小括号）在代数中用于把项组合在一起。它们告诉我们括号外面的运算要作用于括号内的每一项。这是一个关键概念，因为如果你忽略括号或处理不当，你的解就会出错。
Common forms you will see include a(b + c), (a + b)(c + d), or expressions like 2(3x - 4). In linear equations at KS3, we mostly deal with a single set of brackets multiplied by a constant.
你会见到的常见形式包括 a(b + c)、(a + b)(c + d) 或像 2(3x - 4) 这样的表达式。在 KS3 阶段的线性方程中，我们主要处理被一个常数乘的单个括号。
3. Expanding Brackets (Distributive Property) | 展开括号（分配律）
ou multiply the term outside by each term inside the bracket. This is also called expanding the brackets. It is the essential first step when an equation contains brackets.
分配律指出 a(b + c) = a × b + a × c。换句话说，你把括号外面的项乘以括号内的每一项。这也叫做展开括号。当方程含有括号时，这是必不可少的第一步。
For example, expand 4(2x + 3): multiply 4 by 2x to get 8x, and multiply 4 by 3 to get 12. So 4(2x + 3) becomes 8x + 12. Always pay attention to the sign in front of the bracket; a negative multiplier changes the signs of all terms inside.
例如，展开 4(2x + 3)：用 4 乘 2x 得到 8x，再用 4 乘 3 得到 12。因此 4(2x + 3) 变成 8x + 12。务必注意括号前面的符号；负乘数会改变括号内所有项的符号。
4. Equation Solving Steps | 解方程的步骤
Once the brackets are expanded, the equation becomes a standard linear equation. The general steps are: (1) Expand any brackets using the distributive property. (2) Simplify both sides by collecting like terms. (3) Move variable terms to one side and constant terms to the other using inverse operations. (4) Isolate the variable by division or multiplication. (5) Check your solution by substituting it back into the original equation.
一旦括号被展开，方程就变成了一个标准的线性方程。一般步骤是：（1）用分配律展开所有括号。（2）合并同类项，化简方程两边。（3）运用逆运算将含变量项移到一边，常数项移到另一边。（4）通过除法或乘法分离变量。（5）将解代入原方程进行检验。
This sequence works for almost any linear equation at this level. The key is to be systematic and perform the same operation on both sides to keep the equation balanced.
这个顺序几乎适用于该水平的所有线性方程。关键是要有条不紊，并对等式两边执行相同的操作，以保持方程平衡。
5. Example 1: Simple Brackets | 例1：简单括号
8. Finally divide both sides by 2: x = 4. Check: 2(4 + 5) = 2 × 9 = 18, which is correct.
解 2(x + 5) = 18。首先展开括号：2x + 10 = 18。然后两边同时减去 10：2x = 8。最后两边同时除以 2：x = 4。检验：2(4 + 5) = 2 × 9 = 18，正确。
Notice how the bracket is removed immediately. Never try to divide first without expanding, unless the entire bracket is already isolated and the number outside is a factor of the other side. In this case, expanding is safer and clearer.
注意括号是如何被立即去掉的。除非整个括号已经被单独分离且外面的数是另一边的因数，否则切勿在展开括号前先进行除法。在这种情况下，先展开更安全、更清晰。
6. Example 2: Negative Coefficients | 例2：负系数
ides: -2y = -4. Divide by -2: y = 2. Check: 3 - 2(2 - 4) = 3 - 2(-2) = 3 + 4 = 7. Correct.
考虑方程 3 - 2(y - 4) = 7。括号外的乘数是 -2。展开得：3 + (-2)(y) + (-2)(-4) = 7，即 3 - 2y + 8 = 7。化简左边：11 - 2y = 7。两边减 11：-2y = -4。除以 -2：y = 2。检验：3 - 2(2 - 4) = 3 - 2(-2) = 3 + 4 = 7。正确。
A common mistake is to forget that the negative sign applies to the whole bracket. Always write the expanded terms carefully, and double-check the signs when a negative number multiplies a negative term.
一个常见错误是忘记负号作用于整个括号。务必仔细写出展开后的各项，并在负数乘以负项时反复检查符号。
7. Example 3: Variables on Both Sides | 例3：两边含变量
ide by 5: x = 2. Check left: 4(4 - 1) = 4 × 3 = 12; right: 3(2 + 2) = 3 × 4 = 12. Correct.
解方程 4(2x - 1) = 3(x + 2)。首先展开两边：8x - 4 = 3x + 6。接下来，两边同时减 3x，将含变量项移到左边：5x - 4 = 6。然后两边加 4：5x = 10。除以 5：x = 2。检验左边：4(4 - 1) = 4 × 3 = 12；右边：3(2 + 2) = 3 × 4 = 12。正确。
When variables appear on both sides, always aim to collect them on the side where the coefficient is larger to avoid negative coefficients at the end, though it is not strictly necessary.
当方程两边都出现变量时，总是尽量将变量收集到系数更大的一边，以避免最后得到负系数，尽管这并非严格必要。
8. Equations with More than One Bracket | 含有多个括号的方程
2 + 5x - 10 = 14. Combine like terms: 11x - 8 = 14. Add 8: 11x = 22. Divide by 11: x = 2.
有时一个方程在同一边或两边包含两组括号。方法是一样的：分别展开每一组括号，然后化简。例如，2(3x + 1) + 5(x - 2) = 14。展开：6x + 2 + 5x - 10 = 14。合并同类项：11x - 8 = 14。加 8：11x = 22。除以 11：x = 2。
Always be systematic: expand, then collect like terms, then isolate the variable. Rushing through these steps often leads to sign errors or forgetting to multiply a term.
始终要有条不紊：先展开，再合并同类项，然后分离变量。仓促完成这些步骤常常会导致符号错误或漏乘某一项。
9. Checking Your Solution | 检验你的解
Checking is not just a formality; it confirms that your answer satisfies the original equation. Substitute your solution into the original equation, including the brackets exactly as they were. Calculate the left-hand side and the right-hand side separately. If they match, your answer is correct.
检验不仅仅是一种形式，它可以确认你的答案是否满足原方程。将你的解代入原方程，包括原来的括号，严格保持原样。分别计算左边和右边。如果它们相等，你的答案就是正确的。
If the sides do not match, re-check your expansion and your inverse operations. It is much better to find your own mistake than to lose marks in an exam.
如果两边不相等，重新检查你的展开和逆运算。自己发现错误远比在考试中丢分好得多。
10. Common Mistakes to Avoid | 需要避免的常见错误
One common error is forgetting to multiply the constant term inside the bracket. For 3(2x - 5), some students write 6x - 5 instead of 6x - 15. Another mistake is mishandling negative signs: -(x - 3) expands to -x + 3, not -x - 3.
一个常见错误是忘记乘括号内的常数项。对于 3(2x - 5)，有些学生会写成 6x - 5 而不是 6x - 15。另一个错误是负号处理不当：-(x - 3) 展开为 -x + 3，而不是 -x - 3。
A third mistake is dividing before expanding when the bracket is part of a sum or difference. Only divide first if the bracket is completely isolated on one side and the number outside is a factor of the other side. Even then, expanding is usually safer.
第三个错误是当括号是加或减的一部分时，在展开之前先进行除法运算。只有当括号被完全单独放在一边且外面的数是另一边的因数时，才可先进行除法。即便如此，展开通常更为安全。
11. Practice Questions | 练习题
7 - 2(x + 3) = 1; (e) 4(x - 1) + 3(2x + 5) = 31. Expand carefully and check each solution.
尝试解以下方程来检验你的理解。（a）3(x + 4) = 27；（b）5(2x - 3) = 25；（c）2(3x + 1) = 4(x - 2)；（d）7 - 2(x + 3) = 1；（e）4(x - 1) + 3(2x + 5) = 31。仔细展开并检验每一个解。
got a different answer, go back and look for the mistake. Correct work builds confidence.
答案：（a）x=5，（b）x=4，（c）x=-2，（d）x=0，（e）x=2。如果你得到了不同的答案，请回过去找出错误。正确的解题过程能建立信心。
12. Real-life Applications | 实际应用
ere n is the number of people, p is the food price per person, and s is the service charge per person. Solving such an equation helps you find the unknown price per person.
含有括号的方程出现在许多实际问题中。例如，计算团体用餐的总费用，其中包含固定的服务费：总费用 = n(p + s)，n 是人数，p 是每人餐费，s 是每人服务费。解这样的方程可以帮助你找出未知的每人餐费。
2)+(2x-1)) = 2(3x+1) = 6x+2. If the perimeter is given, you can find x using these skills.
另一个例子是计算边长分别为 (x+2) 和 (2x-1) 的矩形的周长。周长公式 2(l + w) 得到 2((x+2)+(2x-1)) = 2(3x+1) = 6x+2。如果已知周长，你就可以运用这些技能求出 x。
Mastering this topic gives you the foundation for more advanced algebra, including solving quadratic equations and rearranging formulas in science.
掌握这一主题将为你学习更高级的代数奠定基础，包括解二次方程以及在科学学科中变换公式。
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