Solving Linear Equations | 解一元一次方程

📚 Solving Linear Equations | 解一元一次方程

Linear equations are the foundation of algebra at Key Stage 3. They allow us to find the value of an unknown number, often represented by a letter such as x, by keeping the equation balanced. This article explains step-by-step methods for solving one-step, two-step, and multi-step equations using inverse operations, brackets, and variables on both sides.

一元一次方程是 KS3 代数的基础。它们能让我们找到一个未知数的值（通常用字母如 x 表示），同时保持等式两边平衡。本文将逐步讲解如何使用逆运算、处理括号以及含有两边变量的方程，一步步解出一步、两步和多步方程。

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1. What Is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal, using the equals sign (=). For example, x + 3 = 7 means that when you add 3 to an unknown number x, you get 7. The aim is to find the value of x 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。例如，x + 3 = 7 表示当你用一个未知数 x 加上 3，得到 7。我们的目标是找出使该陈述成立的 x 的值。

Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.

可以把方程想象成一个平衡的天平。无论你对一边做了什么，必须对另一边做同样的操作，以保持平衡。

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2. One-Step Equations (Addition and Subtraction) | 一步方程（加法和减法）

To solve an equation like x + 4 = 10, we use the inverse operation. The inverse of adding 4 is subtracting 4. So, subtract 4 from both sides: x + 4 − 4 = 10 − 4, which simplifies to x = 6.

要解像 x + 4 = 10 这样的方程，我们使用逆运算。加 4 的逆运算是减 4。所以，两边同时减去 4：x + 4 − 4 = 10 − 4，化简得 x = 6。

For x − 5 = 3, the inverse of subtracting 5 is adding 5. Add 5 to both sides: x − 5 + 5 = 3 + 5, so x = 8.

对于 x − 5 = 3，减 5 的逆运算是加 5。两边同时加上 5：x − 5 + 5 = 3 + 5，因此 x = 8。

Always check your answer by substituting it back into the original equation.

始终通过将答案代回原方程检验。

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3. One-Step Equations (Multiplication and Division) | 一步方程（乘法和除法）

Solving 3x = 12 means ‘3 times x equals 12’. The inverse operation is division by 3. Divide both sides by 3: (3x)/3 = 12/3, giving x = 4.

解 3x = 12 的意思是“3 乘 x 等于 12”。逆运算是除以 3。两边同时除以 3：(3x)/3 = 12/3，得到 x = 4。

For x/5 = 2, the inverse is multiplying by 5. Multiply both sides by 5: (x/5) × 5 = 2 × 5, so x = 10.

对于 x/5 = 2，逆运算是乘以 5。两边同时乘以 5：(x/5) × 5 = 2 × 5，因此 x = 10。

Remember, the fraction line means division, so x/5 is x divided by 5.

记住，分数线表示除法，所以 x/5 是 x 除以 5。

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4. Two-Step Equations | 两步方程

Two-step equations involve two operations. For example, 2x + 3 = 11. First, undo the addition or subtraction, then undo the multiplication or division. Start by subtracting 3 from both sides: 2x + 3 − 3 = 11 − 3 → 2x = 8. Then divide both sides by 2: x = 4.

两步方程包含两种运算。例如，2x + 3 = 11。首先消除加法和减法，然后消除乘法或除法。先两边减 3：2x + 3 − 3 = 11 − 3 → 2x = 8。然后两边除以 2：x = 4。

Another example: x/4 − 1 = 3. Add 1 to both sides: x/4 = 4. Then multiply both sides by 4: x = 16.

另一个例子：x/4 − 1 = 3。两边加 1：x/4 = 4。然后两边乘以 4：x = 16。

Always work in reverse order of operations (undo addition/subtraction before multiplication/division).

始终按运算顺序的逆序操作（先消加法/减法，再消乘法/除法）。

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5. Equations with Brackets | 带括号的方程

If brackets appear, expand them first using the distributive law. For 3(x + 2) = 15, multiply out: 3x + 6 = 15. Then solve as a two-step equation: subtract 6 from both sides (3x = 9), then divide by 3 (x = 3).

如果出现括号，先用分配律展开。对于 3(x + 2) = 15，展开得 3x + 6 = 15。然后按两步方程解：两边减 6（3x = 9），再除以 3（x = 3）。

Sometimes it is easier to divide both sides by the factor outside the bracket first. For 2(x − 4) = 10, you could divide both sides by 2 to get x − 4 = 5, then add 4 to get x = 9. Choose the method that seems quicker.

有时先两边除以括号外的因数会更简单。对于 2(x − 4) = 10，你可以先两边除以 2，得到 x − 4 = 5，然后加 4 得到 x = 9。选择看起来更快的方法。

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6. Equations with Variables on Both Sides | 两边都有变量的方程

When equations have x on both sides, like 5x + 2 = 3x + 8, collect the x terms on one side and the numbers on the other. Subtract 3x from both sides: 5x − 3x + 2 = 8 → 2x + 2 = 8. Then subtract 2 from both sides: 2x = 6, so x = 3.

当方程两边都有 x 时，如 5x + 2 = 3x + 8，把 x 项移到一边，数字移到另一边。两边减去 3x：5x − 3x + 2 = 8 → 2x + 2 = 8。然后两边减 2：2x = 6，所以 x = 3。

If the x terms are subtracted, like 7x − 4 = 2x + 6, subtract 2x from both sides: 5x − 4 = 6. Then add 4 to both sides: 5x = 10, x = 2.

如果 x 项被减去，如 7x − 4 = 2x + 6，两边减 2x：5x − 4 = 6。然后两边加 4：5x = 10，x = 2。

Remember to keep the equation balanced by performing the same operation on both sides.

记住对两边执行相同的操作以保持方程平衡。

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7. Solving by Simplifying First | 先化简再求解

Some equations require simplifying each side before solving. For example, 4x + 3 + 2x − 5 = 20. Combine like terms: (4x + 2x) + (3 − 5) = 6x − 2 = 20. Then add 2 to both sides: 6x = 22, so x = 22/6 = 11/3.

有些方程需要先化简两边再求解。例如，4x + 3 + 2x − 5 = 20。合并同类项：(4x + 2x) + (3 − 5) = 6x − 2 = 20。然后两边加 2：6x = 22，因此 x = 22/6 = 11/3。

Always tidy up each side by collecting like terms and removing unnecessary brackets before applying inverse operations.

在使用逆运算之前，始终通过合并同类项和去掉不必要的括号来整理每一侧。

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8. Equations with Fractional Coefficients | 带分数系数的方程

Equations like (2x)/3 + 1 = 5 can be solved by clearing the fraction first. Multiply every term by the denominator, 3: 3 × (2x)/3 + 3 × 1 = 3 × 5 → 2x + 3 = 15. Then solve: 2x = 12, x = 6.

像 (2x)/3 + 1 = 5 这样的方程可以通过先消去分母来解。每一项都乘以分母 3：3 × (2x)/3 + 3 × 1 = 3 × 5 → 2x + 3 = 15。然后解得：2x = 12，x = 6。

If the equation has more than one fraction, multiply by the least common multiple (LCM) of the denominators. For x/2 + x/3 = 5, the LCM of 2 and 3 is 6. Multiply through by 6: 3x + 2x = 30 → 5x = 30, x = 6.

如果方程有多个分数，乘以所有分母的最小公倍数（LCM）。对于 x/2 + x/3 = 5，2 和 3 的 LCM 是 6。两边乘以 6：3x + 2x = 30 → 5x = 30，x = 6。

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9. Solving Word Problems Using Equations | 用方程解应用题

Many real-life problems can be modelled with linear equations. Start by letting the unknown quantity be x. Write an equation based on the information given, then solve it. Always check that the answer makes sense in the context.

许多实际生活中的问题都可以用一元一次方程来建模。首先设未知量为 x。根据所给信息列方程，然后求解。始终检验答案在情境中是否合理。

Example: ‘Three more than twice a number is 19.’ Let the number be x. Twice the number is 2x, and three more is 2x + 3. The equation is 2x + 3 = 19. Subtract 3: 2x = 16, x = 8.

例子：“一个数的两倍再加三等于 19。”设这个数为 x。两倍是 2x，再加三是 2x + 3。方程为 2x + 3 = 19。减去 3：2x = 16，x = 8。

Practice turning sentences into equations: ‘five less than half a number is 7’ becomes x/2 − 5 = 7.

练习将语句转化为方程：“一个数的一半减去五等于 7”变为 x/2 − 5 = 7。

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10. Common Mistakes to Avoid | 常见错误及避免方法

Mistake 1: Forgetting to apply an operation to both sides. If you subtract 4 from the left, you must also subtract 4 from the right.

错误一：忘记对两边执行相同操作。如果你从左边减去 4，右边也必须减去 4。

Mistake 2: Incorrectly expanding brackets. 3(x + 2) is 3x + 6, not 3x + 2.

错误二：错误地展开括号。3(x + 2) 是 3x + 6，不是 3x + 2。

Mistake 3: Losing track of negative signs. When subtracting a negative term, it becomes addition. For 5x − (−2) = 10, it’s 5x + 2 = 10.

错误三：忽略负号。当减去一个负数项时，变为加法。对于 5x − (−2) = 10，变成 5x + 2 = 10。

Mistake 4: Not checking the solution. Substituting your x back into the original equation will confirm if you are correct.

错误四：不验证解。将 x 代回原方程会确认你是否正确。

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11. Practice Problems | 练习题

Try these equations to test your understanding. Solve each one and then check your answer.

试试这些方程来检验你的理解。解出每一个然后验证答案。

• x + 9 = 15 (Answer: x = 6)

  x + 9 = 15（答案：x = 6）

• 4x = 28 (Answer: x = 7)

  4x = 28（答案：x = 7）

• 2x + 5 = 17 (Answer: x = 6)

  2x + 5 = 17（答案：x = 6）

• 5(x − 3) = 20 (Answer: x = 7)

  5(x − 3) = 20（答案：x = 7）

• 8x − 3 = 2x + 15 (Answer: x = 3)

  8x − 3 = 2x + 15（答案：x = 3）

• x/3 + 4 = 6 (Answer: x = 6)

  x/3 + 4 = 6（答案：x = 6）

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12. Summary | 总结

To solve any linear equation, use inverse operations to isolate the variable while keeping the equation balanced. Tackle one step at a time, simplify where possible, and always check your answer. With practice, these skills become second nature, preparing you for more advanced algebra.

要解任何一元一次方程，使用逆运算分离变量，同时保持方程平衡。一步一步地处理，尽可能化简，并且始终验证你的答案。通过练习，这些技能会变得驾轻就熟，为进一步学习代数做好准备。

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