📚 Year 8 Mathematics: Solving Linear Equations and Inequalities | 8年级数学：解线性方程与不等式
Linear equations are the gateway to algebra. In real life, we use them to calculate costs, distances, speeds, and even to solve puzzles. This article will guide you through the essential skills of solving linear equations and inequalities, step by step, with clear examples and common traps to avoid.
线性方程是代数学习的入门钥匙。在现实生活中，我们用它们计算费用、距离、速度，甚至解决谜题。这篇文章将一步一步带你掌握解线性方程与不等式的基本技巧，配合清晰的例题以及常见易错点，帮助你扎实理解考点。
1. What is a Linear Equation? | 什么是线性方程？
A linear equation is an equation where the unknown variable has a power of 1. For example, 2x + 3 = 11 is linear, but x² + 1 = 10 is not linear because the variable is squared.
线性方程是指未知数的最高次数为1的方程。例如，2x + 3 = 11 是线性的，而 x² + 1 = 10 不是线性的，因为未知数带有平方。
The graph of a linear equation is always a straight line, which is why we call it “linear”. In Year 8, we mainly focus on solving equations where the variable appears only once or twice.
一次方程的图像总是一条直线，所以被称为”线性”。在8年级，我们主要集中解决未知数只出现一次或两次的方程。
- Linear equation: 4x – 7 = 9
- Linear inequality: 3x + 2 ≥ 8
- 线性方程：4x – 7 = 9
- 线性不等式：3x + 2 ≥ 8
2. The Golden Rule: Balancing Equations | 黄金法则：保持等式平衡
An equation is like a balanced seesaw. Whatever you do to one side, you must do the same to the other side. This is the most important rule in solving equations.
等式就像一架平衡的跷跷板。你对一边做了任何操作，都必须对另一边做同样的操作。这是解方程最重要的法则。
For example, to solve x + 5 = 12, subtract 5 from both sides:
例如，解 x + 5 = 12 时，两边同时减去5：
x + 5 – 5 = 12 – 5
x = 7
Always check your answer by substituting back into the original equation:
一定要将答案代入原方程进行检验：
7 + 5 = 12 ✔
3. Solving One-Step Equations | 解一步方程
One-step equations require only one inverse operation. The four basic operations are addition, subtraction, multiplication, and division.
一步方程只需要进行一步逆运算。四种基本运算是加法、减法、乘法和除法。
- Addition: x – 3 = 10 → add 3 to both sides → x = 13
- Subtraction: x + 6 = 15 → subtract 6 from both sides → x = 9
- Multiplication: x ÷ 4 = 2 → multiply both sides by 4 → x = 8
- Division: 5x = 35 → divide both sides by 5 → x = 7
- 加法：x – 3 = 10 → 两边同时加3 → x = 13
- 减法：x + 6 = 15 → 两边同时减6 → x = 9
- 乘法：x ÷ 4 = 2 → 两边同时乘4 → x = 8
- 除法：5x = 35 → 两边同时除以5 → x = 7
When solving, always state the inverse operation clearly. This helps you avoid mistakes and shows your working for marks.
求解时，要清楚写出逆运算。这既能帮助你避免错误，也能展示解题过程以获得步骤分。
4. Solving Two-Step Equations | 解两步方程
Two-step equations require two inverse operations, usually in a specific order: first deal with addition/subtraction, then multiplication/division.
两步方程需要进行两步逆运算，通常顺序是：先处理加减法，再处理乘除法。
Solve 3x + 4 = 19:
解 3x + 4 = 19：
3x + 4 – 4 = 19 – 4 → 3x = 15
3x ÷ 3 = 15 ÷ 3 → x = 5
Check: 3(5) + 4 = 15 + 4 = 19 ✔
检验：3(5) + 4 = 15 + 4 = 19 ✔
Remember: if the equation has a fraction coefficient, for example x/2 + 3 = 7, first subtract 3, then multiply by 2.
记住：如果方程中含有分数系数，例如 x/2 + 3 = 7，先减3，再乘2。
5. Equations with Unknowns on Both Sides | 未知数在方程两边的方程
When variables appear on both sides of the equation, collect them onto one side first. Usually it is easier to make the variable coefficient positive.
当未知数出现在方程两边时，需要先把它们集中到一边。通常让未知数系数为正数会更容易处理。
Solve 5x – 2 = 3x + 8:
解 5x – 2 = 3x + 8：
5x – 3x – 2 = 3x – 3x + 8
2x – 2 = 8
2x = 10 → x = 5
Check: 5(5) – 2 = 23 and 3(5) + 8 = 23 ✔
检验：5(5) – 2 = 23，且 3(5) + 8 = 23 ✔
Always simplify both sides first if possible, including combining like terms and removing brackets.
如果可能，总是先化简方程两边，包括合并同类项和去括号。
6. Expanding Brackets | 展开括号
Brackets appear constantly in equations. To remove them, multiply each term inside the bracket by the term outside.
括号在方程中经常出现。去掉括号的方法是：用括号外的项乘以括号内的每一项。
Expand 3(x + 4):
展开 3(x + 4)：
3(x + 4) = 3 × x + 3 × 4 = 3x + 12
With a negative multiplier, be careful with signs:
遇到负数系数时要特别小心符号：
-2(3x – 5) = -2 × 3x + (-2) × (-5) = -6x + 10
A common mistake is forgetting to multiply the second term. Always check that every inside term is multiplied.
一个常见错误是忘记乘第二个项。务必检查括号内每一项都被乘到了。
7. Solving Equations with Brackets | 解含括号的方程
When an equation has brackets, expand them first, then use the techniques you have learned.
当方程含有括号时，先展开括号，然后运用你已经学会的技巧。
Solve 4(2x – 1) = 3x + 16:
解 4(2x – 1) = 3x + 16：
8x – 4 = 3x + 16
8x – 3x = 16 + 4
5x = 20 → x = 4
Check: 4(2×4 – 1) = 4(7) = 28 and 3×4 + 16 = 12 + 16 = 28 ✔
检验：4(2×4 – 1) = 4(7) = 28，且 3×4 + 16 = 12 + 16 = 28 ✔
If there are brackets on both sides, expand both sides before collecting like terms.
如果方程两边都有括号，先展开两边，再合并同类项。
8. Solving Linear Inequalities | 解线性不等式
Inequalities are solved like equations, with one critical difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
不等式与方程解法类似，但有一个关键区别：如果两边同时乘以或除以一个负数，必须改变不等号的方向。
Solve 5x – 3 > 12:
解 5x – 3 > 12：
5x > 15 → x > 3
Now solve -2x ≤ 8:
现在解 -2x ≤ 8：
x ≥ -4 (sign flipped because we divided by -2)
x ≥ -4（因为除以-2，不等号方向反转）
Why? Consider -2x ≤ 8. If x = -4, -2(-4) = 8, which works. If x = -3, -2(-3) = 6, also works. So x can be -4 or anything greater, hence ≥.
为什么？考虑 -2x ≤ 8。若 x = -4，-2(-4) = 8，成立。若 x = -3，-2(-3) = 6，也成立。因此 x 可以是 -4 或更大的数，即 ≥。
9. Showing Inequalities on a Number Line | 在数轴上表示不等式
A number line helps us visualise the solutions of an inequality. An open circle means the number is not included, while a closed circle means it is included.
数轴可以帮助我们直观地看到不等式的解集。空心圆圈表示该数不在解集内，实心圆圈表示包含该数。
| x > 3 | Open circle at 3, arrow pointing right |
| x ≤ -1 | Closed circle at -1, arrow pointing left |
| x > 3 | 3处画空心圆，箭头向右 |
| x ≤ -1 | -1处画实心圆，箭头向左 |
Remember: < and > use open circles; ≤ and ≥ use closed circles.
记住：< 和 > 用空心圆；≤ 和 ≥ 用实心圆。
10. Word Problems and Common Mistakes | 应用题与常见错误
To solve a word problem, start by defining the unknown variable. Then translate the words into an equation or inequality, solve it, and finally answer the original question.
解决应用题时，先定义未知数，然后将文字翻译成方程或不等式，求解，最后回答原问题。
Example: “A number multiplied by 4 and increased by 7 equals 31.” Let x be the number.
例：”一个数乘以4再加7等于31。”设这个数为 x。
4x + 7 = 31 → 4x = 24 → x = 6
Common mistakes to avoid:
需要避免的常见错误：
- Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
- Adding/ subtracting on only one side of the equation.
- Not expanding brackets correctly, especially with negative signs.
- Not checking the solution in the original equation.
- 乘以或除以负数时忘记反转不等号。
- 只在方程一边做加减运算。
- 去括号时出错，尤其是负号的情况。
- 没有把解代回原方程进行检验。
The more you practise, the fewer mistakes you will make. Always show your working line by line.
练习越多，错误越少。一定要一步一步写出你的解题过程。
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