2.1.2
Jab hum algebraic expressions ko combine karte hain, toh hum sirf unhi terms ko seedha add ya subtract kar sakte hain jinka variable part bilkul same ho. Yeh algebraic simplification ki buniyaad hai aur yeh equations solve karne se lekar calculus tak har jagah milti hai.
Intuition Hum 3x + 5y kyun add nahi kar sakte?Imagine karo ki x aur y alag-alag currencies hain: dollars aur euros. Tum "3 dollars + 5 dollars = 8 dollars" count kar sakte ho (like terms), lekin "3 dollars + 5 euros" alag hi rahenge jab tak ek currency mein convert na karo. Like terms ka same "currency" (variable part) hota hai.
Terms like terms kehlaate hain agar unka variable part identical ho (same letters same powers par). Sirf coefficients (aage ke numbers) alag ho sakte hain.
Examples:
3 x 2 − 7 x 2 x 2 4 ab 9 ab ab 5 x 5 y unlike hain (alag variables)2 x 2 2 x 3 unlike hain (alag powers)
Arithmetic ka distributive law yahan se shuru karo:
a ⋅ c + b ⋅ c = ( a + b ) ⋅ c
Algebra mein apply karo:
3 x + 5 x = ( 3 + 5 ) ⋅ x = 8 x
Variable x x
Lekin unlike terms ke saath:
3 x + 5 y = 3 x + 5 y aage simplify nahi kar sakte .
Worked example Example 1: Simple combinationSimplify karo: 7 x + 3 x − 2 x
Solution:
Teeno terms ka variable part x x 1
Step 1: Like terms identify karo
Sab like terms hain: 7 x 3 x − 2 x 7 + 3 − 2 = 8
WHY this step? Hum count kar rahe hain: "x x x x
Step 3: Variable part lagao
= 8 x
Answer: 8 x
Worked example Example 2: Mixed termsSimplify karo: 5 x 2 + 3 x − 2 x 2 + 7 x − 4
Solution:
Step 1: Like terms group karo (variable part ke hisaab se)
x 2 5 x 2 − 2 x 2 x 3 x 7 x Constant terms: − 4
WHY group? Hum sirf identical variable part wale terms ko combine kar sakte hain. Grouping se yeh clearly dikhta hai.
Step 2: Har group combine karo
x 2 5 x 2 − 2 x 2 = ( 5 − 2 ) x 2 = 3 x 2 x 3 x + 7 x = ( 3 + 7 ) x = 10 x Constants: − 4
WHY alag rakho? x 2 x
Step 3: Standard form mein likho (descending powers)
= 3 x 2 + 10 x − 4
Answer: 3 x 2 + 10 x − 4
Worked example Example 3: Multiple variablesSimplify karo: 4 ab + 2 a + 3 ab − 5 b + a
Solution:
Step 1: Groups identify karo
ab 4 ab 3 ab a 2 a a 1 a b − 5 b WHY ab a b Variable part exactly match karna chahiye. ab a b a b ab 4 ab + 3 ab = 7 ab a 2 a + 1 a = 3 a b − 5 b
Step 3: Final expression likho
= 7 ab + 3 a − 5 b
Answer: 7 ab + 3 a − 5 b
Worked example Example 4: Powers matter!Simplify karo: 6 x 3 + 2 x 2 − 3 x 3 + 5 x 2 + x 3
Solution:
Step 1: Power ke hisaab se alag karo
x 3 6 x 3 − 3 x 3 x 3 x 2 2 x 2 5 x 2 WHY alag karo? x 3 = x ⋅ x ⋅ x x 2 = x ⋅ x
Step 2: Har power combine karo
x 3 6 − 3 + 1 = 4 4 x 3 x 2 2 + 5 = 7 7 x 2 = 4 x 3 + 7 x 2
Answer: 4 x 3 + 7 x 2
Common mistake Mistake 2: Coefficients AUR variables dono add karnaWrong: 4 x + 3 x = 7 x 2 STEEL-MAN: Yeh kyun hota hai?
Hum coefficients sahi add karte hain: 4 + 3 = 7
Variables ko combine hote dekhte hain aur sochte hain "ye aur strong ho jayenge" ya "double ho gaye"
Addition aur multiplication mein confusion: ( x ) ( x ) = x 2
THE FIX: Like terms add karte waqt, variable part wahi rehta hai .
Correct: 4 x + 3 x = 7 x
Socho: "4 seb + 3 seb = 7 seb" (7 seb-squared nahi!)
Recall Feynman Test: 12-saal ke bachche ko samjhaoImagine karo tum fruits collect kar rahe ho. Tumhare paas 3 seb aur 5 seb hain — tum inhe combine kar ke 8 seb pa sakte ho kyunki dono same type ke fruit hain.
Lekin agar tumhare paas 3 seb aur 5 santare hain, tum nahi keh sakte ki tumhare paas "8 seb-santare" hain ya kuch aisa strange. Tumhare paas bas 3 seb aur 5 santare alag-alag hain.
Algebra mein, x y 3 x 5 x x 3 x + 5 x = 8 x 3 x 5 y 3 x + 5 y
Rule: Sirf unhi terms ko combine karo jinka letter-part exactly same ho!
Mnemonic Memory Aid: "SAVE the variable"S ame variable part → A dd the numbers → V ariable stays → E xpress the result
Ya socho: "Like meets Like" — sirf identical variable parts hi combine ho sakte hain.
#flashcards/maths
Like terms kya hote hain? :: Terms jinke variable parts identical hon (same letters same powers par), sirf coefficients mein fark ho
Kya 5 x 2 3 x 3 Nahi, yeh unlike terms hain kyunki x x 2 x 3
Simplify karo: 7 a + 3 b − 2 a + 5 b 5 a + 8 b 7 a − 2 a = 5 a 3 b + 5 b = 8 b
Like terms combine karte waqt kya same rehta hai? Variable part unchanged rehta hai; sirf coefficients add/subtract hote hain
4 x + 3 y Kyunki x y
Simplify karo: 5 x y + 2 x + 3 x y − x 8 x y + x 5 x y + 3 x y = 8 x y 2 x − x = x
− 3 x 2 + 7 x 2 4 4 x 2 − 3 + 7 = 4
Standard form descending powers
