Solving Two-Step Equations | 求解两步方程

📚 Solving Two-Step Equations | 求解两步方程

hem builds the foundation for solving more complex linear equations and real-life problems in KS3 mathematics.

两步方程是需要恰好两次逆运算才能解出未知数的代数等式。它们通常把加减法与乘除法组合在一起，例如 2x + 3 = 11。掌握两步方程是解决更复杂线性方程和现实问题的基础，也是剑桥中学数学的关键考点。

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1. What Are Two-Step Equations? | 什么是两步方程？

7 = 18 is a two-step equation because you need to add 7 and then divide by 5 to isolate x.

两步方程指需要两种不同运算才能求出变量值的方程。通常变量先被乘或除，再加上或减去一个常数。比如 5x − 7 = 18 就是一个两步方程，需要先加 7 再除以 5 才能得到 x。

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2. The Balance Method | 天平平衡法

In mathematics, an equation is like a balanced scale: whatever you do to one side, you must also do to the other side to keep it balanced. This principle is called the balance method. When solving two-step equations, every operation you apply must be performed on both sides of the equals sign to maintain equality.

在数学中，方程就像一个平衡的天平：你对一边做什么，就必须对另一边做同样的事，才能保持平衡。这个原理称为平衡法。解两步方程时，每一步运算都要在等号两边同时进行，以维持等式成立。

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3. Undoing Addition or Subtraction | 逆加减运算

it to both sides. For example, in 4y + 9 = 29, subtract 9 from both sides to get 4y = 20.

一定要先逆运算加减法。如果变量项加了一个数，两边同时减去这个数；如果减去一个数，两边同时加上这个数。比如 4y + 9 = 29，两边同时减去 9，得到 4y = 20。

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4. Undoing Multiplication or Division | 逆乘除运算

by that number. Continuing the example, divide both sides of 4y = 20 by 4 to obtain y = 5.

处理完常数后，通过逆乘除运算把变量单独解出来。如果变量乘以一个数，两边除以这个数；如果变量除以一个数，两边乘以这个数。继续上面的例子，对 4y = 20 两边同时除以 4，得到 y = 5。

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5. Worked Example 1: ax + b = c | 例题1：ax + b = c

Let's solve 3x + 5 = 20 step by step. First, subtract 5 from both sides:

我们来解 3x + 5 = 20，分步进行。先用两边减5：

3x + 5 − 5 = 20 − 5

3x = 15

Next, divide both sides by 3:

接着，两边除以3：

3x ÷ 3 = 15 ÷ 3

x = 5

substituting x = 5 back into the original equation: 3(5) + 5 = 15 + 5 = 20, which is true.

将 x = 5 代入原方程检验：3(5) + 5 = 15 + 5 = 20，成立。

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6. Worked Example 2: x/a + b = c | 例题2：x/a + b = c

Consider the equation x/4 + 3 = 7. Start by subtracting 3 from both sides:

考虑方程 x/4 + 3 = 7。先从两边减去3开始：

x/4 + 3 − 3 = 7 − 3

x/4 = 4

Then multiply both sides by 4 to remove the denominator:

然后两边乘以4，消去分母：

(x/4) × 4 = 4 × 4

x = 16

Substituting x = 16 gives 16/4 + 3 = 4 + 3 = 7, confirming the answer.

代入 x = 16 检验：16/4 + 3 = 4 + 3 = 7，答案正确。

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7. Checking Your Solution | 检验解

Always substitute your found value back into the original equation. This quick check confirms whether the equality holds. If both sides of the equation are equal, the solution is correct. It is a crucial habit that helps avoid careless errors during exams and homework.

每次都要把求出的值代回原方程。这个快速检验能确认等式是否成立。如果两边相等，说明解正确。这是一个关键习惯，能够避免考试和作业中的粗心错误。

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8. Common Mistakes to Avoid | 常见错误

g with negative numbers. For instance, in −2x + 1 = 7, subtract 1 to get −2x = 6, then divide by −2 to get x = −3.

一个常见错误是运算顺序搞错，比如先除后加减。务必先逆加减再逆乘除。另一个错误是忘记对等式两边同时运算。还有要当心符号错误，尤其在涉及负数时。例如 −2x + 1 = 7，先减1得 −2x = 6，再除以 −2 得 x = −3。

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9. Real-World Applications | 实际应用

ed charge plus a rate per kilometer: total cost = 5 + 2x, where x is distance. If you have £ temperature conversions like C = (F − 32) × 5/9 can be rearranged into two-step equations.

两步方程可以模拟许多日常情景。比如出租车费由起步价加每公里收费构成：总费用 = 5 + 2x，x 为距离。如果你有19英镑，解方程 2x + 5 = 19 就能求出最远距离。类似地，温度转换公式 C = (F − 32) × 5/9 也可以变形为两步方程。

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10. Practice Tips and Summary | 练习要点与总结

When practising two-step equations, begin with simple positive coefficients and gradually introduce negatives and fractions. Write each step clearly, showing the balanced operation on both sides. Over time, you will be able to solve them mentally, but always remember the fundamental rule: keep the equation balanced. Regular practice with worksheets and past paper questions will build confidence and speed.

练习两步方程时，先从简单的正系数入手，逐步引入负数和分数。每一步写清楚，标示出两边同时进行的运算。随着练习，你最终能心算求解，但要始终记住基本规则：保持等式平衡。通过练习卷和历年真题定期训练，可以提升信心和速度。

• Always isolate the variable term first.
• Perform the same operation on both sides.
• Check your answer by substitution.

• 总是先分离变量项。
• 两边执行相同运算。
• 用代入法检验答案。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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