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Did you know?
The word "Algebra" comes from the Arabic al-jabr, meaning "the reunion of broken parts". It’s the magical puzzle of mathematics where missing numbers are forced to reveal themselves. In exams like SSC, Banking, and Railways, algebra-based questions appear regularly — approach them like a detective game: find the clues, catch the answer.

A shopkeeper once said: "5 pens and 3 notebooks cost ₹55, while 3 pens and 5 notebooks cost ₹49. Find the price of one pen."
This is the real game of algebra — two facts for two unknowns, then adding and subtracting until the answer emerges.
Variable: Something whose value changes, like *x*, *y*.
Constant: Something fixed, like 5, –3.

🧠 Without looking up: What is the difference between a variable and a constant?

Expression vs Equation — know the trap
2x + 5 is merely an expression — there is no equality sign.
2x + 5 = 13 is an equation — the equal sign has arrived.
This is exactly where exams try to confuse you: an expression cannot be solved; an equation can.

Linear Equation: An equation where the variable’s highest power is 1.
Example: 4x – 7 = 9, or 2y + 3 = 11.
When the power becomes 2, it turns into a quadratic equation — we’ll stick to linear for now.

🧠 *Question: Is 3x² + 2x + 1 = 0 linear? Why or why not?*

Solving a Linear Equation: The Transposition Method
Keep sending numbers to one side until the variable stands alone — and change the sign each time.

+ becomes –, × becomes ÷ when crossing the equals sign.
Example:
5x + 4 = 19
5x = 19 – 4 → 5x = 15 → x = 15 ÷ 5 = 3

Weird Memory Trick (Mnemonic):

"Imagine the equal sign as a magic river.
Numbers must pay a toll — their sign flips!
First move the add/subtract clan, then multiply/divide gets the master plan."

🧠 *Apply it now: Solve 7x – 3 = 18 step by step in your head.*

Simultaneous Linear Equations (Two Variables)
When two equations come with two unknowns, use elimination or substitution.
The trick: Make the coefficients of one variable equal, then subtract.
The pen-notebook riddle:
5x + 3y = 55
3x + 5y = 49
Multiply the first by 3, the second by 5:
15x + 9y = 165
15x + 25y = 245
Subtract: 16y = 80 → y = 5 (notebook), then x = 8 (pen).

🧠 Pause — what was the first key step to solve the two equations above?

Algebraic Identities — Your Shortcut Armoury
These formulas give answers directly, skipping the messy multiplication.

(a + b)² = a² + 2ab + b² — Square the first, square the second, double the product.

(a – b)² = a² – 2ab + b² — Same, only the middle sign changes.

a² – b² = (a + b)(a – b) — Difference of two squares = Sum × Difference.

(x + a)(x + b) = x² + (a + b)x + ab — Middle coefficient = a+b, last term = a×b.

Funny Mnemonic (Make it stick):

"A Boy and a Cat got a square plot.
(a+b)²: First the Boy builds his house (a²), then the Cat builds hers (b²),
and they build two meeting rooms together (2ab).
For (a-b)² the Cat runs away, so the rooms become negative!"
Visualizing this silly image will cement the identities forever.

🧠 Recall the identity for (a – b)² without scrolling up. Whisper it once.

Properties that Make Calculation Easy
Addition and multiplication are commutative (order doesn’t matter), and we can distribute multiplication over addition.
Distributive Law: a(b + c) = ab + ac
This is your mental math superpower: 102 × 5 = (100 + 2)×5 = 500 + 10 = 510. Similarly, 99 × 8 = (100 – 1)×8 = 800 – 8 = 792.

🧠 Use the distributive law to quickly compute 97 × 6.

Word Problems: The Art of Translation
The real exam trap hides inside words.
"Twice a number plus 5 gives 17" → 2x + 5 = 17.
"The difference of two numbers is 4 and their sum is 20" → x – y = 4, x + y = 20.
Thumb Rule: Words like "of", "sum of", "exceeds" convert to mathematical operations.
"more than" → +, "less than" → –, "times" → ×, "divided by" → ÷.

Summary Table: Algebra – Fundamental Toolkit

Review this after 2 days:
Return to this chapter exactly two days later. Spend just 10 minutes — first answer the 🧠 prompts without peeking, then chant the identities aloud. Remember, algebra is detective work — the more you practise, the sharper your catch.