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. A linear equation is an equation where the unknown variable appears only to the first power, such as 2x + 3 = 7. The goal is to find the value of the variable that makes the equation true. Mastering linear equations gives you a powerful tool for solving problems in mathematics and science.

一元一次方程是代数的基础。一元一次方程是指未知数只出现一次幂的方程，例如 2x + 3 = 7。我们的目标是找出使等式成立的未知数值。掌握一元一次方程能为你提供解决数学和科学问题的强大工具。

1. What is an Equation? | 什么是方程？

An equation is a mathematical statement that shows two expressions are equal. It always contains an equals sign (=). The expressions on both sides must balance. Think of an equation as a set of scales – whatever you do to one side, you must do to the other to keep it balanced.

方程是表明两个表达式相等的数学陈述。它总是包含一个等号（=）。两边的表达式必须平衡。把方程想象成天平——你对一边做了什么，必须对另一边做同样的操作，才能保持平衡。

2. The Balance Method | 天平法

The balance method is the key idea for solving equations. To keep the scales balanced, add, subtract, multiply or divide both sides by the same number. For x + 5 = 12, subtract 5 from both sides: x + 5 − 5 = 12 − 5, so x = 7.

天平法是解方程的核心思想。为保持天平平衡，两边同时加、减、乘、除同一个数。对于 x + 5 = 12，两边同时减去 5：x + 5 − 5 = 12 − 5，得到 x = 7。

3. Solving by Adding or Subtracting | 通过加减解方程

When a number is added to or subtracted from the variable, use the inverse operation. For x − 4 = 9, add 4 to both sides: x − 4 + 4 = 9 + 4, giving x = 13. For y + 8 = 3, subtract 8 from both sides: y + 8 − 8 = 3 − 8, so y = −5.

当未知数加上或减去一个数时，使用逆运算。对于 x − 4 = 9，两边同时加 4：x − 4 + 4 = 9 + 4，得到 x = 13。对于 y + 8 = 3，两边同时减 8：y + 8 − 8 = 3 − 8，所以 y = −5。

4. Solving by Multiplying or Dividing | 通过乘除解方程

If the variable is multiplied by a coefficient, divide both sides by that coefficient. For 5p = 30, divide both sides by 5: 5p ÷ 5 = 30 ÷ 5, so p = 6. If the equation has a fraction with the variable in the numerator, multiply both sides by the denominator. For x/4 = −3, multiply both sides by 4: (x/4) × 4 = −3 × 4, so x = −12.

如果未知数乘以一个系数，两边同时除以该系数。对于 5p = 30，两边同除以 5：5p ÷ 5 = 30 ÷ 5，得到 p = 6。如果方程含有未知数在分子的分数，两边乘分母。对于 x/4 = −3，两边乘 4：(x/4) × 4 = −3 × 4，得到 x = −12。

5. Two-step Equations | 两步方程

Many linear equations require two steps. For 2x + 3 = 11, first subtract 3 from both sides: 2x = 8. Then divide both sides by 2: x = 4. Always undo addition or subtraction before undoing multiplication or division – reverse the order of operations.

许多一元一次方程需要两步。对于 2x + 3 = 11，首先两边减 3：2x = 8。然后两边除以 2：x = 4。总是先消去加减法，再消去乘除法——即逆用运算顺序。

6. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law. For 3(y − 2) = 15, multiply out: 3y − 6 = 15. Then add 6 to both sides: 3y = 21. Finally divide by 3: y = 7.

当方程含有括号时，先用分配律展开。对于 3(y − 2) = 15，展开得 3y − 6 = 15。然后两边加 6：3y = 21。最后除以 3：y = 7。

7. Variables on Both Sides | 两边都含未知数

If the variable appears on both sides, collect all variable terms on one side and constants on the other. For 5m + 2 = 3m + 10, subtract 3m from both sides: 2m + 2 = 10. Then subtract 2: 2m = 8, so m = 4.

如果未知数出现在两边，把所有含未知数的项移到一边，常数项移到另一边。对于 5m + 2 = 3m + 10，两边减 3m 得到 2m + 2 = 10。然后减 2：2m = 8，所以 m = 4。

8. Equations with Fractions | 含分数的方程

To solve equations with fractions, find a common denominator and multiply every term by it to clear the fractions. For (x/2) + (1/3) = 2, multiply every term by 6: 3x + 2 = 12. Then subtract 2: 3x = 10, so x = 10/3.

解含有分数的方程时，先找出公分母，乘以每一项消去分母。对于 (x/2) + (1/3) = 2，每一项乘 6 得 3x + 2 = 12。然后减 2：3x = 10，所以 x = 10/3。

9. Word Problems Leading to Linear Equations | 引出一元一次方程的文字题

Many real-life situations can be modelled with linear equations. Let the unknown quantity be x, then form an equation from the given information. For example: “The sum of three consecutive numbers is 57. Find the numbers.” Let the numbers be n, n+1, n+2. Then n + (n+1) + (n+2) = 57 → 3n + 3 = 57 → 3n = 54 → n = 18. The numbers are 18, 19, 20.

许多现实问题可以用一元一次方程来建模。设未知量为 x，再根据已知信息列出方程。例如：“三个连续整数的和为 57，求这三个数。”设这三个数为 n, n+1, n+2。则 n + (n+1) + (n+2) = 57 → 3n + 3 = 57 → 3n = 54 → n = 18。这三个数是 18, 19, 20。

10. Checking Your Answers | 检查答案

Always substitute your solution back into the original equation to check it is correct. For 2(3x − 1) = 4x + 6, if you find x = 4, check: left side = 2(3×4 − 1) = 2(12 − 1) = 22; right side = 4×4 + 6 = 22. Both sides equal, so the answer is correct.

总是把求得的结果代入原方程检验是否正确。对于 2(3x − 1) = 4x + 6，假设你解得 x = 4，检验：左边 = 2(3×4 − 1) = 2(12 − 1) = 22；右边 = 4×4 + 6 = 22。两边相等，答案正确。

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

Watch out for sign errors when moving terms across the equals sign. Do not forget to multiply every term inside the brackets when expanding. Never divide by zero. When solving 5 = 2 − x, it might help to rewrite as 2 − x = 5, then subtract 2: −x = 3, so x = −3.

移项时注意符号错误。展开括号时不要忘记乘括号内的每一项。永远不要除以零。在解 5 = 2 − x 时，可以先写成 2 − x = 5，然后减 2：−x = 3，所以 x = −3。

12. Practice and Build Fluency | 练习与提升熟练度

Solving linear equations is a skill that improves with practice. Start with simple one-step equations and work up to those with brackets and fractions. Over time, you will develop fluency and be able to solve them mentally. This confidence will help you tackle more advanced topics like simultaneous equations and quadratic equations.

解一元一次方程是一项熟能生巧的技能。从简单的一步方程开始，逐步进阶到含括号和分数的方程。久而久之，你将能够心算并快速求解。这种信心将帮助你应对更高级的主题，如联立方程和二次方程。

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Solving Linear Equations | 解一元一次方程