🗺️ Chapter Roadmap
1 The Building Blocks
Algebra is the branch of mathematics where we use English letters (like x, y, a, b) to represent unknown numbers. Let's understand the basic vocabulary:
A letter used to represent an unknown number. Its value can change (vary).
A fixed number. Its value never changes.
The number attached (multiplied) to a variable. In 4x, the 4 is the coefficient.
2 Expressions vs. Equations
Many students get confused between these two. The difference is very simple: the equal sign (=).
Algebraic Expression
A combination of variables, numbers, and operations (+, -, ×, ÷). It does NOT have an equal sign.
Algebraic Equation
A mathematical statement that two expressions are equal. It MUST have an equal sign.
3 Magic Formulas (Identities)
These 3 formulas are the backbone of Algebra. You must memorize them for any competitive exam.
- 1 (a + b)2 = a2 + 2ab + b2
- 2 (a - b)2 = a2 - 2ab + b2
- 3 a2 - b2 = (a + b)(a - b)
💡 Solved Example using Formula 1:
Expand: (x + 3)2
(x)2 + 2(x)(3) + (3)2
= x2 + 6x + 9
4 Solving Linear Equations
An equation is like a weighing scale. To keep it balanced, whatever you do to one side, you MUST do to the other side. Our goal is to leave the variable (x) alone on one side.
If a number is + on one side, it becomes - when moved to the other side.
If a number is × on one side, it becomes ÷ when moved to the other side.
📝 Step-by-Step Example:
-
1
Move the +5 to the other side. It becomes -5.
3x = 20 - 5 → 3x = 15 -
2
The 3 is multiplied by x. Move it to the other side to Divide.
x = 153 → x = 5
5 Translating English to Math
In exams, questions are often asked in English sentences. You have to translate them into mathematical equations.
| English Phrase | Math Operation | Example |
|---|---|---|
| Sum, increased by, more than | + | A number increased by 5 → x + 5 |
| Difference, decreased by, less than | - | 5 less than a number → x - 5 |
| Product, times, twice | × | Twice a number → 2x |
| Is, gives, yields, results in | = | Twice a number is 10 → 2x = 10 |