Linear equations in one variable — solving, transposition method

2.1.7 · Maths › Algebra — Introduction & Intermediate

What Is a Linear Equation in One Variable?

"Linear" kyun? $y=ax+b$ ka graph ek seedhi line hoti hai. Solution wahan hota hai jahan yeh line $c$ height par ek horizontal line ko cross karti hai.

"One variable" kyun? Dhundne ke liye sirf ek letter hota hai. Agar hamare paas $2x+3y=7$ hota, toh woh do variables hote — alag game hai.

The Transposition Method: Step-by-Step

Goal: $x$ ko ek side pe akela kar do taaki hum $x=\text{something}$ paayein.

Yeh kaam kyun karta hai? Hum actually dono sides par inverse operation perform kar rahe hain:

• $x+5=12$ → dono sides se $5$ subtract karo → $x=12-5$
• "$+5$ doosri side pe $-5$ ban ke chala jaata hai" likhna shortcut hai

Chalo transposition rules ko scratch se derive karte hain.

Derivation from First Principles

Starting point: Ek equation equality ka ek statement hoti hai. Golden rule: dono sides ke saath ek jaisi cheez karo.

Case 1: Addition/Subtraction

Start: $x+a=b$

Inverse operation apply karo (dono sides se $a$ subtract karo): $x+a-a=b-a$ $x=b-a$

Shortcut: "left side ka $+a$ right side pe $-a$ ban jaata hai."

Case 2: Multiplication/Division

Start: $ax=b$ (jahan $a\ne 0$)

Inverse operation apply karo (dono sides ko $a$ se divide karo): $\frac{ax}{a}=\frac{b}{a}$ $x=\frac{b}{a}$

Shortcut: "left side ka $\times a$ right side pe $÷a$ ban jaata hai."

Agar dono hon toh? Operations ko PEMDAS ke reverse order mein undo karo. Agar equation "pehle multiply phir add" se bani hai, toh hum "pehle subtract phir divide" se undo karte hain.

Worked Examples

Standard Algorithm

4-Step Process:

1. Simplify karo har side ko alag se (like terms combine karo, brackets expand karo)
2. Collect karo saare variable terms ek side, constants doosri side (zaroorat padne par transpose karo)
3. Simplify karo taaki $ax=b$ mile
4. Divide karo dono sides ko coefficient se: $x=\frac{b}{a}$

Order matters: Hamesha addition/subtraction pehle undo karo phir multiplication/division (PEMDAS ka reverse).

Common Mistakes

Key Formulas & Facts

Connections

• 2.1.01-Variables-and-constants — $x$ aur $a$ ka kya matlab hai
• 2.1.06-Forming-equations-from-word-problems — yeh equations kahaan se aati hain
• 2.1.08-Linear-equations-in-two-variables — next level: do unknowns
• 2.3.01-Linear-inequalities — agar $=$ ki jagah $<$ ho toh?
• 2.5.02-Balancing-chemical-equations — chemistry mein wahi transposition logic
• 3.2.04-Graphical-solution-of-linear-equations — geometric view

Practice Recall

#flashcards/maths

Linear equation in one variable kya hoti hai? :: Ek equation jahan variable sirf first power par aata hai, $ax+b=c$ ya equivalent form mein, jahan $a\ne 0$.

Multiplication ke liye transposition rule batao. :: Jab ek coefficient variable ko multiply karta hai, toh woh division ke roop mein cross karta hai (aur division multiplication ke roop mein cross karta hai).

Concept Map