Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the backbone of algebra and appear everywhere in mathematics. In this article, we will explore what linear equations are, how to solve them step by step, and how to avoid common mistakes. You will also see real-life situations where these skills come in handy. By the end, you should feel confident tackling one‑step, two‑step, and even more challenging equations.

线性方程是代数的基石，在数学中随处可见。本文将介绍什么是线性方程，如何一步步求解，以及如何避免常见错误。你还会看到这些技能在生活中的实际应用。学完本文后，你将有信心解决一步方程、两步方程，甚至更复杂的方程。

1. What is a Linear Equation? | 什么是线性方程？

A linear equation is an equation that forms a straight line when graphed. The highest power of the variable (often called x) is 1. For example, 2x + 3 = 7 and x − 5 = 0 are linear equations. They always involve an equals sign, showing that two expressions have the same value.

线性方程是指图像为一条直线的方程。变量（通常称为 x）的最高次数为 1。例如，2x + 3 = 7 和 x − 5 = 0 都是线性方程。它们始终包含等号，表示两个表达式具有相同的值。

In KS3, we only deal with equations in one variable, which means we have only one unknown to find. The solution is the value that makes the equation true. For x + 2 = 5, the solution is x = 3 because 3 + 2 = 5.

在 KS3 阶段，我们只处理一元方程，即只有一个未知数需要求解。解就是使方程成立的数值。对于 x + 2 = 5，解是 x = 3，因为 3 + 2 = 5。

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2. Key Components: Variables, Constants, Coefficients | 关键要素：变量、常数、系数

Every linear equation contains a few important parts. The variable is the unknown, often written as x, y, or another letter. The constant is a number on its own, like 3 or −7. The coefficient is the number multiplied by the variable, for instance the 2 in 2x.

每个线性方程都包含几个重要部分。变量是未知数，通常写作 x、y 或其他字母。常数是单独的数字，例如 3 或 −7。系数是与变量相乘的数字，例如 2x 中的 2。

Understanding these terms helps you describe equations clearly. For the equation 4n − 9 = 15, the variable is n, the coefficient is 4, and the constants are −9 and 15. Spotting these quickly makes solving much easier.

理解这些术语有助于你清晰地描述方程。对于方程 4n − 9 = 15，变量是 n，系数是 4，常数是 −9 和 15。快速识别它们能使求解容易得多。

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3. The Balance Method | 天平法（平衡法）

The most common way to solve linear equations is the balance method. Think of an equation as a pair of balanced scales. Whatever you do to one side, you must do exactly the same to the other side to keep them balanced. This idea is the golden rule of equation solving.

解线性方程最常用的方法是天平法（平衡法）。把方程想象成一架平衡的天平。你对一侧所做的任何操作，必须对另一侧做完全相同的操作，才能保持平衡。这个思想是解方程的黄金法则。

For example, to solve x + 4 = 10, we subtract 4 from both sides: x + 4 − 4 = 10 − 4, giving x = 6. The scales stay balanced because we removed the same weight from each side.

例如，要解 x + 4 = 10，我们从两边同时减去 4：x + 4 − 4 = 10 − 4，得到 x = 6。天平保持平衡，因为我们从每侧移除了相同的重量。

Always aim to isolate the variable on one side. Use inverse operations: addition undoes subtraction, multiplication undoes division, and vice versa.

始终力求将变量单独留在等式的一侧。使用逆运算：加法取消减法，乘法取消除法，反之亦然。

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4. Solving One‑Step Equations | 解一步方程

One‑step equations require just one inverse operation to find the solution. They are the simplest type. For instance, x − 3 = 8 is solved by adding 3 to both sides: x = 11. The operation was subtraction, so we used addition.

一步方程只需一次逆运算即可求解，是最简单的类型。例如，x − 3 = 8 可通过在两边加 3 来解：x = 11。原来的运算是减法，所以我们使用了加法。

Another example is 5x = 20. Since x is multiplied by 5, we divide both sides by 5: x = 4. Similarly, x ÷ 7 = 2 is solved by multiplying both sides by 7, giving x = 14.

另一个例子是 5x = 20。因为 x 与 5 相乘，我们将两边除以 5：x = 4。同样，x ÷ 7 = 2 可通过两边乘以 7 来解，得到 x = 14。

• If the equation is x + a = b, subtract a from both sides.
• If the equation is x − a = b, add a to both sides.
• If the equation is a x = b, divide both sides by a.
• If the equation is x ÷ a = b, multiply both sides by a.

• 如果方程是 x + a = b，两边减 a。
• 如果方程是 x − a = b，两边加 a。
• 如果方程是 a x = b，两边除以 a。
• 如果方程是 x ÷ a = b，两边乘以 a。

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5. Solving Two‑Step Equations | 解两步方程

Two‑step equations involve two operations, so we undo them in reverse order according to BIDMAS. For example, 2x + 5 = 13. First, subtract 5 from both sides (undoing the addition): 2x = 8. Then divide by 2 (undoing the multiplication): x = 4.

两步方程包含两种运算，因此我们按照 BIDMAS 的逆序来撤销它们。例如，2x + 5 = 13。首先两边减 5（撤销加法）：2x = 8。然后除以 2（撤销乘法）：x = 4。

Another example: 7x − 3 = 18. Add 3 to both sides first: 7x = 21. Then divide by 7: x = 3. Always work backwards from the order you would normally calculate an expression.

另一个例子：7x − 3 = 18。先两边加 3：7x = 21。然后除以 7：x = 3。始终从正常计算表达式的顺序反向操作。

Sometimes the variable term looks like x/4 + 6 = 10. Subtract 6 first: x/4 = 4. Then multiply by 4: x = 16. Step‑by‑step thinking prevents errors.

有时变量项形如 x/4 + 6 = 10。先减 6：x/4 = 4。然后乘以 4：x = 16。逐步思考能防止错误。

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6. Equations with Brackets | 带括号的方程

When brackets appear, always expand them first using the distributive law. For example, 3(x + 2) = 15. Multiply out the bracket: 3 × x + 3 × 2 = 3x + 6. So the equation becomes 3x + 6 = 15. Then solve as a two‑step equation: subtract 6, divide by 3, giving x = 3.

当出现括号时，始终先用分配律展开括号。例如，3(x + 2) = 15。将括号乘开：3 × x + 3 × 2 = 3x + 6。于是方程变为 3x + 6 = 15。然后作为两步方程求解：减 6，除以 3，得到 x = 3。

For 2(4x − 3) = 14, expand to 8x − 6 = 14. Add 6 to both sides: 8x = 20. Divide by 8: x = 2.5. Brackets are just a way of grouping, and expanding removes them.

对于 2(4x − 3) = 14，展开得 8x − 6 = 14。两边加 6：8x = 20。除以 8：x = 2.5。括号只是一种分组方式，展开后即可消除。

Never try to divide first without expanding unless the whole bracket is multiplied by a number and you treat it as a single term. Expanding is safer at KS3.

除非整个括号作为一个整体与一个数相乘，否则不要在没有展开括号的情况下先进行除法。在 KS3 阶段，展开括号更安全。

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7. Equations with Variables on Both Sides | 变量在两侧的方程

When x appears on both sides of the equation, we need to collect all variable terms on one side and constants on the other. For example, 5x + 2 = 3x + 10. Subtract 3x from both sides: 2x + 2 = 10. Then subtract 2: 2x = 8. Divide by 2: x = 4.

当 x 出现在等号两侧时，我们需要将所有含变量的项移到一侧，所有常数移到另一侧。例如，5x + 2 = 3x + 10。两边减 3x：2x + 2 = 10。再减 2：2x = 8。除以 2：x = 4。

Another example: 4x − 7 = x + 5. Subtract x from both sides: 3x − 7 = 5. Add 7: 3x = 12. Divide by 3: x = 4. Always aim to make the coefficient of x positive.

另一个例子：4x − 7 = x + 5。两边减 x：3x − 7 = 5。加 7：3x = 12。除以 3：x = 4。始终力求使 x 的系数为正数。

If a negative x appears, like 10 − 2x = 3x, you can add 2x to both sides: 10 = 5x, then x = 2. This keeps calculations simple.

如果出现负的 x，例如 10 − 2x = 3x，你可以两边加 2x：10 = 5x，然后 x = 2。这样可以保持计算简单。

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8. Checking Your Solution | 检验你的解

After finding a solution, always substitute it back into the original equation to make sure it works. For 2x + 5 = 13, we found x = 4. Left side: 2(4) + 5 = 8 + 5 = 13. Right side: 13. Both sides match, so the solution is correct.

找到解后，务必将其代回原方程以确保成立。对于 2x + 5 = 13，我们得到 x = 4。左边：2(4) + 5 = 8 + 5 = 13。右边：13。两边相等，因此解是正确的。

This check is especially useful when an equation has brackets or variables on both sides. If the left and right sides do not give the same number, you have made a mistake somewhere. Re‑working with the check in mind is a powerful habit.

当方程含有括号或两侧都有变量时，这种检验尤其有用。如果左边和右边得到的数不相等，就说明某处出现了错误。带着检验意识重新计算是一个强大的习惯。

In exams, doing a quick mental check can save you from losing easy marks. It only takes a few seconds but confirms your answer is logical.

在考试中，快速心算检验可以避免丢失容易得到的分数。这只需几秒钟，但能确认你的答案合乎逻辑。

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9. Common Mistakes to Avoid | 常见错误要避免

Many students forget to apply an operation to both sides. For example, in x/3 = 5, multiplying only the left side by 3 gives x = 5, which is wrong. You must multiply the right side as well: x/3 × 3 = 5 × 3 → x = 15.

许多学生忘记对两边同时进行运算。例如，在 x/3 = 5 中，仅将左边乘以 3 会得到 x = 5，这是错误的。你必须也将右边乘以 3：x/3 × 3 = 5 × 3 → x = 15。

Another error is mishandling negative signs. In 2 − 3x = 8, students might incorrectly move the 2. The correct step: subtract 2 first, giving −3x = 6, then divide by −3 to get x = −2. Always write each step neatly.

另一个错误是负号处理不当。在 2 − 3x = 8 中，学生可能会错误地移动 2。正确的步骤：先减 2，得到 −3x = 6，然后除以 −3 得到 x = −2。始终整洁地书写每一步。

Also, when expanding brackets, remember to multiply every term inside. 2(x + 3) is 2x + 6, not 2x + 3. This is perhaps the most common slip‑up at KS3.

此外，展开括号时，记得乘以括号内的每一项。2(x + 3) 是 2x + 6，而不是 2x + 3。这可能是 KS3 阶段最常见的疏漏。

Avoid dividing before collecting like terms. Keep the balance in mind and work methodically.

避免在合并同类项之前进行除法。心中牢记平衡，有条不紊地运算。

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10. Real‑Life Applications | 实际应用

Linear equations model many everyday situations. For instance, if a taxi charges £3.50 flag‑down plus £1.20 per kilometre, the cost C for a journey of d km is C = 1.2d + 3.5. To find the distance when the cost is £10.10, solve 1.2d + 3.5 = 10.1.

线性方程可以用来模拟许多日常情形。例如，如果一辆出租车起步价 £3.50，之后每公里 £1.20，行驶 d 公里的费用 C 为 C = 1.2d + 3.5。要找出费用为 £10.10 时的距离，需要解方程 1.2d + 3.5 = 10.1。

Subtract 3.5: 1.2d = 6.6. Divide by 1.2: d = 5.5 km. Equations help you plan budgets, calculate cooking times, or adjust recipes. They are a powerful problem‑solving tool.

减 3.5：1.2d = 6.6。除以 1.2：d = 5.5 公里。方程可帮你规划预算、计算烹饪时间或调整食谱。它们是强有力的解题工具。

In science, you might use F = ma (force equals mass times acceleration) and solve for mass or acceleration. The ability to rearrange and solve linear equations is vital across many subjects.

在科学中，你可能会使用 F = ma（力等于质量乘以加速度），并求解质量或加速度。重新排列并求解线性方程的能力在众多学科中都至关重要。

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11. Summary: Steps to Success | 总结：成功步骤

1. Simplify both sides – expand brackets and collect like terms if needed.
2. Move all variable terms to one side and constants to the other using inverse operations.
3. Perform the same operation on both sides to maintain balance.
4. Isolate the variable by dividing by the coefficient.
5. Check your solution by substituting it back into the original equation.

1. 化简两边——必要时展开括号并合并同类项。
2. 利用逆运算将所有含变量的项移至一侧，常数移至另一侧。
3. 对两边进行相同的运算，以保持平衡。
4. 通过除以系数将变量分离出来。
5. 将解代回原方程进行检验。

Mastering linear equations opens the door to more advanced algebra, graphs, and problem solving. Practice with a variety of equations—including those with fractions and decimals—and soon you will solve them with confidence.

掌握线性方程会为你打开通往更高级代数、图像和问题解决的大门。用各种各样的方程进行练习——包括含有分数和小数的方程——很快你就能充满自信地求解它们。

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