📚 Solving Linear Equations with Brackets and Unknowns on Both Sides | 解含括号和两边含未知数的一元一次方程
0. You will learn how to expand, collect like terms, move unknowns, and check your final answer step by step.
在 KS3 数学中，解一元一次方程是你需要掌握的最重要的代数技能之一。本文重点讲解含有括号且两边都含未知数的方程，例如 3(x + 2) = 2x + 10。你将学习如何展开括号、合并同类项、移动未知数，并一步步检验最终答案。
1. What Is a Linear Equation? | 什么是线性方程？
+ 5 = 13 and 4(x − 1) = x + 7 are both linear equations because x is not squared or cubed.
线性方程是指未知数（通常写作 x）只以一次方出现的方程。例如 2x + 5 = 13 和 4(x − 1) = x + 7 都是线性方程，因为 x 没有被平方或立方。
The goal when solving a linear equation is to find the value of x that makes the left-hand side equal to the right-hand side. This value is called the solution or root of the equation.
解一元一次方程的目标是找到使左边等于右边的 x 值。这个值被称为方程的解或根。
2. Expanding Brackets First | 先展开括号
When an equation contains brackets, your first step is usually to expand them using the distributive law. For example, 2(x + 3) means 2 multiplied by every term inside the bracket.
当方程含有括号时，第一步通常是使用分配律展开括号。例如 2(x + 3) 表示 2 乘以括号内的每一项。
s to 2x + 6. Always check that you have multiplied the number outside the bracket by every term inside it.
这得到 2 × x + 2 × 3，化简为 2x + 6。一定要检查你是否用括号外的数乘以了括号内的每一项。
2(x + 3) = 2x + 6
3. Collecting Like Terms | 合并同类项
Like terms are terms that contain exactly the same variable raised to the same power. For example, 5x and −2x are like terms, while 5x and 5 are not like terms.
同类项是指含有完全相同变量且指数相同的项。例如 5x 和 −2x 是同类项，而 5x 和 5 不是同类项。
After expanding brackets, you should collect all x terms together and all number terms together on each side of the equation. This makes the equation much easier to solve.
展开括号后，应在方程每一边将所有含 x 的项合并在一起，将所有数字项合并在一起。这会使方程更容易求解。
4. Moving Unknowns to One Side | 把未知数移到一边
all x terms to one side. This is done by adding or subtracting the same x term from both sides.
如果方程两边都含 x 项，例如 3x + 2 = x + 10，你需要把所有的 x 项移到一边。方法是在方程两边同时加上或减去同一个 x 项。
of 3x + 2 = x + 10 to get 2x + 2 = 10. Now the unknown appears only on the left-hand side.
例如，在 3x + 2 = x + 10 两边同时减去 x，得到 2x + 2 = 10。现在未知数只出现在左边。
3x + 2 = x + 10 → 3x − x + 2 = 10 → 2x + 2 = 10
5. Solving Step by Step | 分步求解
xpand the bracket to get 2x + 6 = x + 8. Then subtract x from both sides to get x + 6 = 8.
我们来解方程 2(x + 3) = x + 8。首先展开括号得到 2x + 6 = x + 8。然后两边同时减去 x，得到 x + 6 = 8。
ives x = 2. Always perform the same operation on both sides to keep the equation balanced.
最后两边同时减去 6，将 x 单独分离出来。得到 x = 2。一定要对两边进行相同的运算，以保持方程平衡。
2(x + 3) = x + 8 → 2x + 6 = x + 8 → x + 6 = 8 → x = 2
6. Checking Your Answer | 检验答案
x + 3) = x + 8, the left-hand side is 2(2 + 3) = 10 and the right-hand side is 2 + 8 = 10.
求出解后，应将其代回原方程检验。对于 2(x + 3) = x + 8 中的 x = 2，左边为 2(2 + 3) = 10，右边为 2 + 8 = 10。
ect. If the sides are not equal, you should review your expansion or your moving of terms.
由于两边相等，所以解 x = 2 是正确的。如果两边不相等，你应该重新检查括号展开或移项的过程。
7. Common Mistakes to Avoid | 常见错误
One common mistake is forgetting to multiply every term inside the bracket. For example, 3(x + 4) is 3x + 12, not 3x + 4.
一个常见的错误是忘记乘以括号内的每一项。例如 3(x + 4) 等于 3x + 12，而不是 3x + 4。
Another mistake is changing the sign incorrectly when moving terms across the equals sign. Remember that subtracting a negative term is the same as adding its positive value.
另一个错误是在将项移到等号另一边时符号变化不正确。记住，减去一个负数项等同于加上它的正值。
Also, do not combine x terms with number terms. The expression 5x + 3 cannot be simplified to 8x or 8, because they are not like terms.
此外，不要把 x 项与数字项合并。5x + 3 不能化简为 8x 或 8，因为它们不是同类项。
8. Word Problems and Forming Equations | 应用题与列方程
ree times a number plus 4 equals twice the number plus 10, you can write 3x + 4 = 2x + 10.
许多实际问题可以通过建立一元一次方程来解决。例如，如果一个数的 3 倍加 4 等于这个数的 2 倍加 10，你可以写成 3x + 4 = 2x + 10。
btract 2x from both sides to get x + 4 = 10, so x = 6. This means the unknown number is 6.
两边同时减去 2x，得到 x + 4 = 10，因此 x = 6。这意味着未知数是 6。
Always define your variable clearly at the start. Write "Let x be the unknown number" before you form the equation.
开始时应明确定义变量。在列方程之前，先写“设 x 为未知数”。
9. Using a Table to Organise Steps | 用表格整理步骤
tions with brackets and unknowns on both sides. Below is an example for 4(x − 2) = 3x + 1.
在解含括号且两边含未知数的方程时，表格可以帮助你记录每一步的操作。下面是 4(x − 2) = 3x + 1 的例子。
| Step | 步骤 | Equation | 方程 |
| Expand brackets | 展开括号 | 4x − 8 = 3x + 1 |
| Subtract 3x from both sides | 两边减 3x | x − 8 = 1 |
| Add 8 to both sides | 两边加 8 | x = 9 |
Check: 4(9 − 2) = 4 × 7 = 28 and 3 × 9 + 1 = 28, so the solution is correct.
检验：4(9 − 2) = 4 × 7 = 28，3 × 9 + 1 = 28，所以解是正确的。
10. Graphical Interpretation | 图形解释
the point where two straight lines cross. For example, solving 2x + 1 = x + 5 gives x = 4.
一元一次方程可以从图形上理解为两条直线相交的点。例如，解方程 2x + 1 = x + 5 得到 x = 4。
, they will intersect at the point (4, 9). The x-coordinate of the intersection is the solution of the equation.
如果画出直线 y = 2x + 1 和 y = x + 5，它们会相交于点 (4, 9)。交点的 x 坐标就是方程的解。
2x + 1 = x + 5 → x = 4 → intersection at (4, 9)
11. Extension: Equations with Fractional Coefficients | 拓展：分数系数方程
− 2. To solve this, you can multiply every term by the denominator to clear the fraction.
有时未知数的系数是分数，例如 (1/2)x + 3 = x − 2。要解这个方程，可以将每一项乘以分母以消去分数。
m by 2 to get x + 6 = 2x − 4. Then subtract x from both sides to get 6 = x − 4, so x = 10.
将每一项乘以 2，得到 x + 6 = 2x − 4。然后两边同时减去 x，得到 6 = x − 4，所以 x = 10。
Always remember to multiply both sides of the equation by the same number. This keeps the equation balanced and produces an equivalent equation.
一定要记住在方程的两边同时乘以同一个数。这能保持方程平衡，并得到等价的方程。
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