← 返回资讯屏轩教育 · 行业资讯

Solving Two-Step Linear Equations | 两步线性方程的解法

2026-08-21 12:14 · 本文部分内容由AI辅助创作

📚 Solving Two-Step Linear Equations | 两步线性方程的解法

Linear equations are at the heart of Key Stage 3 algebra. In this revision guide, you will learn how to solve two-step linear equations confidently using inverse operations and the balancing method. You will also see worked examples, common mistakes, word problems and exam tips to build strong foundational skills.

线性方程是 KS3 代数的核心内容。在本复习指南中，你将学习如何利用逆运算和平衡法自信地解两步线性方程。你还将看到例题、常见错误、文字题和考试技巧，以建立扎实的基础技能。

1. What Is a Two-Step Linear Equation? | 什么是两步线性方程

he variable x is first multiplied by 3 and then 4 is added. To isolate x, you must reverse these two steps in the correct order. The highest power of the variable is 1, so the graph of the equation is a straight line. This is why it is called a linear equation.

两步线性方程是指未知量恰好经历两种运算的方程。例如，在方程 3x + 4 = 19 中，变量 x 先乘以 3，然后加上 4。要分离 x，必须按正确顺序逆转这两个步骤。变量最高次数为 1，因此方程图像是一条直线。这就是它被称为线性方程的原因。

divided by 5 first, and then 2 is subtracted. The two operations are division and subtraction. Solving the equation means finding the value of x that makes the left side equal to the right side.

另一个例子是 x/5 − 2 = 6。这里 x 先除以 5，然后再减去 2。这两种运算是除法和减法。解方程就是找出使左边等于右边的 x 值。

2. Key Vocabulary and Inverse Operations | 关键术语与逆运算

Before solving, make sure you know these terms: equation, variable, coefficient, constant, operation, inverse operation and balance. The inverse of addition is subtraction, the inverse of subtraction is addition, the inverse of multiplication is division, and the inverse of division is multiplication.

在解方程之前，请确保掌握这些术语：方程、变量、系数、常数、运算、逆运算和平衡。加法的逆运算是减法，减法的逆运算是加法，乘法的逆运算是除法，除法的逆运算是乘法。

Operation | 运算Inverse Operation | 逆运算
Addition + | 加法 +Subtraction − | 减法 −
Subtraction − | 减法 −Addition + | 加法 +
Multiplication × | 乘法 ×Division ÷ | 除法 ÷
Division ÷ | 除法 ÷Multiplication × | 乘法 ×

Understanding inverse operations is essential because solving an equation is like working backwards through the order of operations. You undo the last operation first, then the first operation.

理解逆运算至关重要，因为解方程就像逆向执行运算顺序。你先撤销最后一步运算，再撤销第一步运算。

3. The Balancing Method | 平衡法

Think of an equation as a balanced set of scales. Whatever you do to one side, you must do to the other side to keep it balanced. This is the golden rule of solving equations. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side.

把方程想象成一组平衡的天平。无论你对一边做什么，都必须对另一边做同样的操作，以保持平衡。这是解方程的黄金法则。如果你对一边加、减、乘或除以一个数，你必须对另一边做完全相同的操作。

For example, if you subtract 5 from the left side, you must subtract 5 from the right side too. If you divide the left side by 4, you must divide the right side by 4. Keeping the equation balanced ensures that the equality remains true.

例如，如果你从左边减去 5，那么右边也必须减去 5。如果你将左边除以 4，那么右边也必须除以 4。保持方程平衡可以确保等式仍然成立。

This method is often called the balance method because it keeps both sides equal, just like keeping two pans of a scale at the same height.

这种方法通常称为平衡法，因为它使两边保持相等，就像使天平的两个托盘保持在同一高度一样。

广告 / Sponsored

海外直订Higher Mathematics for AQA GCSE 高等数学AQA GCSE
海外直订Higher Mathematics for AQA GCSE 高等数学AQA GCSE
¥645.0
📱 去京东购买
京东精选

4. Solving ax + b = c | 解 ax + b = c 型方程

In this form, the variable is first multiplied by a number and then another number is added. To solve, reverse the order: subtract the constant first, then divide by the coefficient.

在这种形式中，变量先乘以一个数，然后再加上另一个数。求解时，要逆向操作：先减去常数，再除以系数。

Worked example 1: Solve 3x + 4 = 19.

例题 1：解方程 3x + 4 = 19。

3x + 4 = 19

Step 1: Subtract 4 from both sides to undo the addition. 3x + 4 − 4 = 19 − 4, so 3x = 15.

第 1 步：两边同时减去 4，抵消加法。3x + 4 − 4 = 19 − 4，得到 3x = 15。

Step 2: Divide both sides by 3 to undo the multiplication. 3x ÷ 3 = 15 ÷ 3, so x = 5.

第 2 步：两边同时除以 3，抵消乘法。3x ÷ 3 = 15 ÷ 3，得到 x = 5。

The solution is x = 5.

方程的解为 x = 5。

Worked example 2: Solve 2x + 5 = 17.

例题 2：解方程 2x + 5 = 17。

2x + 5 = 17

Subtract 5 from both sides: 2x = 12. Then divide both sides by 2: x = 6.

两边同时减去 5：2x = 12。然后两边同时除以 2：x = 6。

Worked example 3: Solve 5x − 3 = 22.

例题 3：解方程 5x − 3 = 22。

5x − 3 = 22

ant is subtracted, so undo it by adding 3 to both sides: 5x = 25. Then divide by 5: x = 5.

这里常数被减去，因此两边同时加上 3 来抵消：5x = 25。然后除以 5：x = 5。

5. Solving x/a + b = c | 解 x/a + b = c 型方程

In this form, the variable is first divided by a number and then another number is added or subtracted. To solve, undo the addition or subtraction first, then undo the division by multiplying.

在这种形式中，变量先除以一个数，然后再加上或减去另一个数。求解时，先抵消加法或减法，再通过乘法抵消除法。

Worked example: Solve x/4 + 6 = 10.

例题：解方程 x/4 + 6 = 10。

x/4 + 6 = 10

Step 1: Subtract 6 from both sides: x/4 = 4.

第 1 步：两边同时减去 6：x/4 = 4。

Step 2: Multiply both sides by 4: x = 16.

第 2 步：两边同时乘以 4：x = 16。

Remember: when the variable is divided by a number, multiply both sides by that number to undo the division.

记住：当变量被一个数除时，两边同时乘以该数以抵消除法。

Worked example 2: Solve x/7 − 2 = 3.

例题 2：解方程 x/7 − 2 = 3。

x/7 − 2 = 3

Add 2 to both sides: x/7 = 5. Then multiply both sides by 7: x = 35.

两边同时加上 2：x/7 = 5。然后两边同时乘以 7：x = 35。

6. Negative Coefficients and Constants | 负系数与负常数

the constant is negative, be extra careful with signs. Worked example: Solve 5 − 2x = 11.

当变量系数或常数为负数时，要格外注意符号。例题：解方程 5 − 2x = 11。

5 − 2x = 11

Step 1: Subtract 5 from both sides to isolate the term containing x: −2x = 6.

第 1 步：两边同时减去 5，分离含 x 的项：−2x = 6。

Step 2: Divide both sides by −2: x = −3.

第 2 步：两边同时除以 −2：x = −3。

Be careful with signs: a negative divided by a negative is positive; a positive divided by a negative is negative. If the coefficient is negative, the solution will have the opposite sign to the constant term.

注意符号：负数除以负数得正；正数除以负数得负。如果系数为负，解的符号将与常数项相反。

Worked example 2: Solve 12 − 4y = 0.

例题 2：解方程 12 − 4y = 0。

Subtract 12 from both sides: −4y = −12. Divide by −4: y = 3.

两边同时减去 12：−4y = −12。除以 −4：y = 3。

7. Checking Your Answer by Substitution | 代入检验答案

19 with x = 5, the left side becomes 3(5) + 4 = 15 + 4 = 19, which equals the right side.

始终将解代回原方程检验。这是检查答案是否正确的最可靠方法。对于 3x + 4 = 19，当 x = 5 时，左边变为 3(5) + 4 = 15 + 4 = 19，等于右边。

If both sides are equal, your answer is correct. If not, go back and check your inverse operations, especially the order of steps and the signs of your numbers.

如果两边相等，说明答案正确。如果不相等，请返回检查你的逆运算，尤其是步骤顺序和数字符号。

: 5 − 2(−3) = 5 + 6 = 11. The left side equals the right side, so the solution is correct.

例如，检验 x = −3 是否满足 5 − 2x = 11，将 −3 代入 x：5 − 2(−3) = 5 + 6 = 11。左边等于右边，因此解是正确的。

8. Common Mistakes to Avoid | 常见错误与避免方法