📚 Introduction to Algebra | 代数入门
Algebra is the branch of mathematics where letters and symbols are used to represent numbers and quantities in formulas and equations. For Year 7 students, algebra marks an exciting step from concrete arithmetic to abstract thinking.
代数是数学的一个分支，它用字母和符号来表示数字、数量以及公式和方程中的关系。对于7年级学生来说，代数标志着从具体算术迈向抽象思维的一次激动人心的跨越。
1. What is Algebra? | 什么是代数？
Algebra is like a puzzle where some numbers are missing. Instead of writing “a number plus 3 equals 10”, we write x + 3 = 10. The letter x stands for the unknown value we need to find.
代数就像一个缺少数字的谜题。我们不再写“一个数加3等于10”，而是写成x + 3 = 10。字母x代表我们需要求解的未知数值。
Letters used in algebra are called variables. They can change or vary, which is why they are called “variables”.
代数中使用的字母称为变量。它们可以变化或改变，因此被称为“变量”。
2. Using Letters for Numbers | 用字母表示数
In arithmetic, we write 5 + 3 = 8. In algebra, we might write n + 3 = 8, where n represents the number we don’t know yet. The letter can be any letter, such as a, b, c, x, y or n.
在算术中，我们写5 + 3 = 8。在代数中，我们可以写n + 3 = 8，其中n代表我们还不知道的数。字母可以是任意字母，如a、b、c、x、y或n。
When we write a number next to a letter, it means multiplication. For example, 5n means 5 × n. We usually don’t write the multiplication sign in algebra to avoid confusion with the letter x.
当我们在字母旁边写一个数时，它表示乘法。例如，5n表示5 × n。在代数中我们通常不写乘号，以免与字母x混淆。
- 3a means 3 × a
- 4xy means 4 × x × y
- a² means a × a
- 3a表示3 × a
- 4xy表示4 × x × y
- a²表示a × a
3. Algebraic Expressions | 代数表达式
An algebraic expression is a combination of numbers, letters and operations (like +, −, ×, ÷). For example, 3x + 5 is an expression. It has two parts: 3x and 5, joined by the + sign.
代数表达式是数字、字母和运算符号（如+、−、×、÷）的组合。例如，3x + 5是一个表达式。它由两部分组成：3x和5，用+号连接。
Each part of an expression is called a term. Terms that have the same variable are called like terms. For example, 2x and 5x are like terms, but 2x and 5y are not like terms.
表达式的每一部分称为项。含有相同变量的项称为同类项。例如，2x和5x是同类项，而2x和5y不是同类项。
4. Simplifying Expressions | 化简表达式
To simplify an expression, we combine like terms. For example, simplify 4x + 3x:
要化简表达式，我们需要合并同类项。例如，化简4x + 3x：
4x + 3x = (4 + 3)x = 7x
We can also simplify expressions with constants. For example, 5x + 2 + 3x + 4 becomes 5x + 3x + 2 + 4 = 8x + 6.
我们也可以合并常数项。例如，5x + 2 + 3x + 4化简为5x + 3x + 2 + 4 = 8x + 6。
| Expression | Simplified |
| 2a + 5a | 7a |
| 3b + 4 + 2b + 1 | 5b + 5 |
| 6c − 2c + 3 | 4c + 3 |
5. Substitution | 代入求值
Substitution means replacing a letter with a number. If we know that x = 4, then the expression 3x + 2 becomes 3 × 4 + 2 = 12 + 2 = 14.
代入就是用数字替换字母。如果我们知道x = 4，那么表达式3x + 2就变成3 × 4 + 2 = 12 + 2 = 14。
When substituting, always follow the order of operations: brackets first, then indices (powers), then multiplication and division, and finally addition and subtraction (BIDMAS).
代入时，务必遵循运算顺序：先括号，再指数（幂），然后乘除，最后加减（BIDMAS）。
If a = 5 and b = 2, then 4a − 3b = 4(5) − 3(2) = 20 − 6 = 14
6. Solving Simple Equations | 解简单方程
An equation shows that two expressions are equal. Solving an equation means finding the value of the variable that makes the equation true. For example, solve x + 7 = 12.
方程表示两个表达式相等。解方程就是找到使方程成立的变量的值。例如，求解x + 7 = 12。
To solve, we do the inverse operation on both sides. The inverse of +7 is −7, so:
求解时，我们在方程两边进行逆运算。+7的逆运算是−7，所以：
x + 7 − 7 = 12 − 7 → x = 5
Check: 5 + 7 = 12 ✓. Always check your answer by substituting it back into the original equation.
检验：5 + 7 = 12 ✓。一定要把答案代回原方程进行检验。
7. Inequalities | 不等式
An inequality compares two values using symbols like: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).
不等式用符号比较两个值：<（小于）、>（大于）、≤（小于或等于）、≥（大于或等于）。
For example, x + 3 > 10 means x + 3 is greater than 10. Subtracting 3 from both sides gives x > 7. So x can be any number larger than 7, such as 8, 9.5, or 100.
例如，x + 3 > 10表示x加3大于10。两边同时减去3，得到x > 7。因此x可以是任何大于7的数，比如8、9.5或100。
Remember: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
记住：如果不等式两边同时乘以或除以一个负数，必须改变不等号的方向。
8. Sequences | 数列
Algebra helps us describe number patterns. A sequence is a list of numbers that follow a rule. For example, 3, 6, 9, 12, … is a sequence of multiples of 3.
代数帮助我们描述数字规律。数列是按一定规则排列的一组数。例如，3, 6, 9, 12, … 是3的倍数组成的数列。
We can write a rule for the n-th term. For the sequence above, the n-th term is 3n. When n = 1, 3n = 3; when n = 2, 3n = 6; and so on.
我们可以写出第n项的表达式。对于上面的数列，第n项是3n。当n = 1时，3n = 3；当n = 2时，3n = 6；以此类推。
Another example: the sequence 5, 7, 9, 11, … increases by 2 each time. Its n-th term is 2n + 3. Check: n = 1 gives 2(1) + 3 = 5 ✓.
另一个例子：数列5, 7, 9, 11, …每次增加2。它的第n项是2n + 3。检验：n = 1时，2(1) + 3 = 5 ✓。
9. Real-World Applications | 实际应用
Algebra is not just for the classroom. It appears everywhere: calculating the cost of items, finding distances, designing buildings, and even in computer programming.
代数不仅仅用于课堂。它无处不在：计算商品价格、求距离、设计建筑，甚至计算机编程中都用得到。
For example, if a taxi charges a fixed fee of $3 plus $2 per kilometre, the total cost C for a journey of d kilometres is C = 3 + 2d. If you travel 8 km, then C = 3 + 2(8) = 3 + 16 = 19 dollars.
例如，如果出租车起步价3美元，每公里2美元，那么行驶d公里的总费用C为C = 3 + 2d。如果行驶8公里，则C = 3 + 2(8) = 3 + 16 = 19美元。
Using algebra allows us to solve problems with unknown quantities quickly and clearly.
使用代数可以帮助我们快速、清晰地解决含有未知量的问题。
10. Common Mistakes to Avoid | 常见错误避免
Here are some common errors that Year 7 students often make in algebra, and how to avoid them.
以下是7年级学生在代数学习中常见的错误以及如何避免它们。
- Mistake: Writing 3x + 2x = 5x².
Correct: 3x + 2x = 5x. You only add the coefficients (3 + 2), not the variable. - Mistake: Mixing unlike terms, e.g., 2x + 3y = 5xy.
Correct: 2x + 3y cannot be simplified because x and y are different variables. - Mistake: Forgetting the order of operations when substituting.
Correct: Always calculate powers and multiplications before additions and subtractions. - Mistake: Dropping the negative sign.
Correct: When simplifying −3x − 2x, the result is −5x, not 5x.
- 错误：写成3x + 2x = 5x²。
正确：3x + 2x = 5x。只需合并系数（3 + 2），变量不变。 - 错误：合并不同类项，如2x + 3y = 5xy。
正确：2x + 3y不能化简，因为x和y是不同的变量。 - 错误：代入时忘记运算顺序。
正确：先算幂和乘法，再算加法和减法。 - 错误：丢掉负号。
正确：化简−3x − 2x时，结果是−5x，而不是5x。
Practice these skills step by step, and algebra will become your friend.
一步一步地练习这些技巧，代数就会成为你的好朋友。
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