Addition and subtraction of algebraic expressions

2.1.3 · Maths › Algebra — Introduction & Intermediate

Overview

Algebraic expressions mathematical phrases hain jo variables, constants, aur operations ko combine karti hain. Inhe add aur subtract karna equations ko manipulate karne aur problems solve karne ki foundation hai—lekin arithmetic ke unlike, hum sirf like terms ko hi combine kar sakte hain.

Figure — Addition and subtraction of algebraic expressions

Core Concepts

The Process: Step-by-Step Derivation

Why This Works: First Principles

Chaliye derive karte hain ki hum like terms ko distributive property use karke kyun add kar sakte hain.

Starting point: Distributive property kehti hai $a (b + c) = ab + a c$

Forward direction (kyun hum combine KAR sakte hain): $3 x + 5 x$

Yeh aslaan mein kya hai? Har term matlab hai " $x$ ki kitni copies":

$3 x$ matlab $x + x + x$
$5 x$ matlab $x + x + x + x$

Toh: $3 x + 5 x = 3 times x + x + x + 5 times x + x + x + x = 8 times x + x + x + x + x + x = 8 x$

Distributive property use karke (reverse): $3 x + 5 x = (3 + 5) x = 8 x$

Yeh step kyun? Hum common $x$ ko "factor out" kar rahe hain, phir coefficients add kar rahe hain.

Backward direction (kyun hum unlike terms combine NAHI kar sakte): $3 x + 5 y$

Factor karne ki koshish karo: Humein ek common factor chahiye hoga, lekin $x$ aur $y$ different variables hain. $3 x + 5 y = (3 \cdot x) + (5 \cdot y)$

Yahan koi common factor nahi hai jo bahar nikala ja sake, isliye yeh $3 x + 5 y$ hi rehta hai. Hum isse aur simplify nahi kar sakte.

Worked Examples

Worked example Example 3: Multiple Variables and Powers

Problem: Simplify $(6 x^{2} y - 3 x y^{2} + 2 x) + (4 x y^{2} - 2 x^{2} y + 5 y) - (x^{2} y - x y^{2} + 3 x)$

Solution:

Step 1: Saare parentheses hataao (signs dekho!) $= 6 x^{2} y - 3 x y^{2} + 2 x + 4 x y^{2} - 2 x^{2} y + 5 y - x^{2} y + x y^{2} - 3 x$

Yeh step kyun? Pehle do sets ke aage $+$ hai, toh signs same rehte hain. Last set ke aage $-$ hai, toh saare signs flip ho jaate hain.

Step 2: Saari like terms identify karo

$x^{2} y$ terms: $6 x^{2} y$ , $- 2 x^{2} y$ , $- x^{2} y$
$x y^{2}$ terms: $- 3 x y^{2}$ , $4 x y^{2}$ , $x y^{2}$
$x$ terms: $2 x$ , $- 3 x$
$y$ terms: $5 y$ Step 3: Har group combine karo

x^2y \text{ terms:} &\quad 6x^2y - 2x^2y - x^2y = (6-2-1)x^2y = 3x^2y \\ xy^2 \text{ terms:} &\quad -3xy^2 + 4xy^2 + xy^2 = (-3+4+1)xy^2 = 2xy^2 \\ x \text{ terms:} &\quad 2x - 3x = (2-3)x = -x \\ y \text{ terms:} &\quad 5y \text{ (koi aur } y \text{ terms nahi)} \end{align}$$ **Step 4:** Final expression likho $$= 3x^2y + 2xy^2 - x + 5y$$ **Answer:** $3x^2y + 2xy^2 - x + 5y$

Common Mistakes & How to Fix Them

Common mistake Mistake 3: Coefficients Ki Jagah Exponents Add Karna

Galat approach: $2 x^{3} + 5 x^{3} = 7 x^{6}$

Yeh sahi kyun lagta hai: Multiply karte waqt hum exponents ADD karte hain: $x^{3} \cdot x^{3} = x^{6}$ . Students rules mix up kar dete hain.

Steel-man: Kuch situations HAIN jahan exponents add hote hain (multiplication) aur kuch jahan coefficients multiply hote hain (woh bhi multiplication: $2 x \cdot 3 x = 6 x^{2}$ ). Genuinely confusing hai ki kaunsa rule kab lagta hai.

Fix:

Addition/subtraction: Sirf coefficients combine hote hain. Exponents same rehte hain. $2 x^{3} + 5 x^{3} = (2 + 5) x^{3} = 7 x^{3}$
Multiplication: Coefficients multiply hote hain, exponents add hote hain. $2 x^{3} \cdot 5 x^{3} = (2 \cdot 5) x^{3 + 3} = 10 x^{6}$

Yaad rakhne ka tarika: Addition matlab "same cheez zyada count karna." Agar tumhare paas $x^{3}$ ke 2 boxes hain aur 5 aur boxes add ho, toh tumhare paas $x^{3}$ ke 7 boxes hain—koi alag power nahi!

Memory Aids

Recall Feynman Explanation (12-saal ke bache ko samjhao)

Socho tum apna toy collection organize kar rahe ho. Tumhare paas hai:

3 red cars
5 blue cars
2 red planes Agar main poochhun "tumhare paas kitni red cars hain jab tumhare dost tumhe 4 aur red cars deta hai?", tum bologe $3 + 4 = 7$ red cars. Easy!

Lekin agar main poochhun "kitne toys hain agar tum 3 red cars aur 5 blue cars combine karo?" Tum nahi bol sakte "8 red-blue cars"—yeh sense nahi banta! Tumhare paas 3 red cars AUR 5 blue cars hain. Woh alag rehte hain kyunki woh different types hain.

Algebra bhi same hai:

$3 x + 4 x = 7 x$ ✓ (same type: dono " $x$ " cheezein hain)
$3 x + 5 y$ stays $3 x + 5 y$ ✗ (different types: " $x$ " cheezein aur " $y$ " cheezein) Letter ( $x$ , $y$ ) batata hai ki tumhare paas kaunsi "type" ki cheez hai. Aage ka number (coefficient) batata hai kitni hain. Tum sirf "kitni" wala part tab add kar sakte ho jab "type" bilkul same ho!

Subtraction signs kyun flip karta hai? Socho tumhara dost toys waapis maangta hai. Agar woh chahta hai ki tum "2 red cars aur 3 blue cars" hataao, tum dono types le rahe ho: $- 2$ red cars aur $- 3$ blue cars. Minus sab par lagta hai jo woh le ja rahe hain!

Practice Problems

Simplify karo: $(8 m + 5 n - 3) + (2 m - 7 n + 9)$
Simplify karo: $(10 p^{2} - 4 p + 6) - (3 p^{2} + 2 p - 5)$
Simplify karo: $(4 a^{2} b - 3 ab + 7) + (- 2 a^{2} b + ab - 2) - (a^{2} b - 5 ab + 3)$
Error analysis: Ek student ne likha $(6 x - 4) - (2 x - 1) = 4 x - 5$ . Unki galti dhundho aur explain karo.

Connections Distributive Property — Like terms combine karne ki foundation

Combining Like Terms — Yeh note hi woh technique hai
Polynomials — Multiple terms wale expressions jinhe hum add/subtract karte hain
Factoring Algebraic Expressions — Reverse process (combining → splitting)
Solving Linear Equations — Variables isolate karne ke liye yeh skills use hoti hain
Order of Operations — Complex expressions mein add/subtract kab karna hai yeh decide karta hai

Flashcards

Like terms kya hote hain? :: Same variables wale terms jo same powers par raised hon. Example: $5 x^{2}$ aur $- 3 x^{2}$ like terms hain; $5 x^{2}$ aur $5 x$ like terms NAHI hain.

Like terms hi add/subtract kyun ho sakte hain?

Distributive property ki wajah se.

3 x + 5 x = (3 + 5) x = 8 x

common

x

factor out karke. Unlike terms mein koi common factor nahi hota jo bahar nikala ja sake.

Parentheses mein likhe expression ko subtract karte waqt signs ka kya hota hai?

Parentheses ke andar ki har term ka sign flip hota hai.

(a - b + c) - (d - e + f) = a - b + c - d + e - f

. Minus

- 1

se multiplication ki tarah distribute hota hai.

Kya

2 x^{2} + 3 x^{2}

simplify ho sakta hai, aur agar haan toh kya milega?

Haan. Dono terms mein

x^{2}

hai, toh woh like terms hain.

2 x^{2} + 3 x^{2} = (2 + 3) x^{2} = 5 x^{2}

Kya

4 x + 7 y

aur simplify ho sakta hai?

Nahi.

x

aur

y

different variables hain, isliye yeh unlike terms hain. Expression

4 x + 7 y

hi rehta hai.

5 x + 3 y = 8 x y

mein kya galti hai?

Unlike terms ko combine nahi kiya ja sakta.

5 x

(paanch

x

's) aur

3 y

(teen

y

's) different types hain, jaise seb aur santre. Expression

5 x + 3 y

hi rehta hai.

(9 a - 4) - (3 a + 2)

simplify karo

Negative distribute karo:

9 a - 4 - 3 a - 2

. Like terms combine karo:

(9 a - 3 a) + (- 4 - 2) = 6 a - 6

Addition/subtraction mein exponents wale terms mein kya combine hota hai: coefficients ya exponents?

Coefficients combine hote hain. Exponents same rehte hain.

4 x^{5} + 7 x^{5} = (4 + 7) x^{5} = 11 x^{5}

, na ki

11 x^{10}