Solving 2×2 systems using Cramer's rule: Foundations

2.6.11 · D1 · Maths › Matrices & Determinants — Introduction › Solving 2×2 systems using Cramer's rule

Yeh page assume karta hai ki tumne kuch nahi dekha. Hum har ek symbol build karenge jo parent note (Cramer's rule) mein aata hai, ek aisi order mein jahan har naya idea sirf pehle waale ideas par rely karta hai.

1. Ek unknown, aur letters $x$ aur $y$

Isse picture karo. $x$ ko ek empty box $□$ ki tarah socho. Equation ek clue hai ki us box mein kaunsa number jaata hai. Jab hum "solve" karte hain, hum woh ek number dhoondh rahe hote hain jo har clue ko sach banata hai.

Yeh topic isko kyun zarurat hai. Cramer's rule exist karta hai do unknowns ek saath dhoondhne ke liye — woh do secret numbers jo do equations ke andar chhupe hain. "Ek letter ek aisi number ke liye khada hai jo hum discover karna chahte hain" — yeh idea ke bina, kuch bhi solve karna possible nahi.

2. Ek linear equation, aur "linear" ka matlab

Isse picture karo — yeh page ki sabse important picture hai. $x$ aur $y$ mein har linear equation ek bilkul seedhi line draw karta hai jab tum un sabhi $(x, y)$ points ko mark karte ho jo ise satisfy karte hain. Isliye ise linear kaha jata hai — iska graph ek line hai.

Figure — Solving 2×2 systems using Cramer's rule

Figure dekho: red line par koi bhi point chuno, uski horizontal position ( $x$ ) aur vertical position ( $y$ ) padho, $x + 2 y = 4$ mein plug karo, aur yeh kaam karta hai. Line se bahar ke points equation mein fail hote hain.

Yeh topic isko kyun zarurat hai. Poora parent note do aisi equations ke baare mein hai. Har ek ek line hai. Answer wahan hai jahan woh milti hain — toh hum pehle ek equation = ek line par agree karna chahte hain.

3. Coefficient — woh number jo unknown ko multiply karta hai

Subscripts padhna. Niche wala chota number — subscript — sirf ek name tag hai, multiplication nahi. $a_{1}$ ka matlab hai "equation 1 se $a$ "; $a_{2}$ ka matlab hai "equation 2 se $a$ ." Yeh do alag fixed numbers hain jo ek hi role play karte hain (dono "coefficient of $x$ " hain) do alag equations mein.

Yeh topic isko kyun zarurat hai. Cramer's rule poori tarah inhi coefficients se bana hai — ek grid mein arrange karke. Agar tum $a_{1}$ (coefficient of $x$ , row 1) ko $b_{1}$ (coefficient of $y$ , row 1) se alag nahi kar sakte, toh grid ka koi matlab nahi.

4. Constant $c$ — daayein taraf ka answer

Isse picture karo. Line figure mein, $c$ control karta hai ki line origin se kitni door baithi hai. $c$ change karo aur line page par slide karti hai, same tilt rakhte hue.

Yeh topic isko kyun zarurat hai. Do constants $c_{1}, c_{2}$ "answer column" banate hain. Cramer ka magic yeh hai ki is column ko coefficient grid mein slide karo — toh hum ise apne aap mein ek cheez ke roop mein jaanna chahte hain.

5. System — do equations, dono ek saath sach

Isse picture karo. Ek sheet of paper par do lines.

Solution woh single point (red dot) hai jo ek saath dono lines par hai — woh akela $(x, y)$ jo dono clues maanta hai.

Yeh topic isko kyun zarurat hai. Yeh crossing point wahi answer hai jo Cramer's rule compute karta hai. Aage ki har cheez is dot ko locate karne ke baare mein hai.

6. Matrix aur coefficient matrix

Isse picture karo — yeh memory mein burn kar lo.

Red column notice karo: column ek variable ko own karta hai. Column 1 $x$ ka hai, column 2 $y$ ka. Yeh ek fact hai jis wajah se Cramer ek column replace karta hai (kabhi row nahi).

Yeh topic isko kyun zarurat hai. Determinant jise hum abhi meet karne wale hain woh is grid ko process karta hai. Aur "column swap" jo har unknown dhoondhta hai woh tabhi sense banta hai jab tum dekho ki $x$ column 1 mein rehta hai.

Dekho bhi Matrix form of linear equations Ax=b, jo poore system ko $A x = c$ mein pack karta hai.

7. Determinant — criss-cross number

Ab parent note ka star symbol.

Yeh exact recipe kyun — "main diagonal minus anti-diagonal"? Down-right jaane wale do numbers multiply karo (main diagonal, $a$ aur $d$ ), phir down-left jaane waale subtract karo (anti-diagonal, $b$ aur $c$ ). Yeh tool ek specific question ka answer deta hai: "Kya is grid ki do rows genuinely alag directions mein point karti hain, ya woh secretly parallel hain?" Jab answer $0$ hota hai, woh parallel hain.

Isse picture karo — woh number kya measure karta hai. Har row ko origin se ek arrow ki tarah treat karo: $(a, b)$ aur $(c, d)$ . Woh do arrows ek parallelogram span karte hain. Determinant us parallelogram ka (signed) area hai, red mein dikhaya gaya. Agar arrows line up ho jaayein, parallelogram flat squash ho jaata hai — zero area — aur $det = 0$ .

Har sign case (taaki tum kabhi surprise na ho):

$a d - b c > 0$ : rows "counter-clockwise" spread hoti hain — positive area.
$a d - b c < 0$ : rows "clockwise" spread hoti hain — signed area negative hai. Bilkul normal; Cramer do aisi signed numbers ko divide karta hai aur signs khud sort ho jaate hain (parent mein Example 1 mein $- 26$ ko $- 14$ se divide kiya gaya hai).
$a d - b c = 0$ : rows parallel — degenerate, area collapse ho jaata hai.

Yeh topic isko kyun zarurat hai. Cramer's rule sirf aur sirf teen aise determinants ke ratios hain. Poora treatment Determinant of a 2×2 matrix mein hai.

8. Division, aur kyun " $D \neq = 0$ " poora game hai

Cramer's rule likhta hai $x = \frac{D _{x}}{D}$ . Bottom $D = det A$ hai. Section 7 se, $D = 0$ exactly tab hota hai jab do lines parallel hain — koi clean crossing point nahi — toh yeh bilkul sense banta hai ki formula wahan answer dene se mana karta hai. Algebra (undefined division) aur geometry (parallel lines) ek hi kahani sunate hain.

Teen outcomes — ek solution, koi nahi, ya infinitely many — Consistency of linear systems se sort hote hain.

9. Notation ko ek saath jodna

Ab parent ke use kiye gaye har ek symbol tumhare paas hai:

Symbol	Plain matlab	Uski picture
$x, y$	do unknown numbers	crossing point ki horizontal & vertical jagah
$a_{1}, b_{1}, \dots$	coefficients (subscript = row)	har line ke tilt/steepness ingredients
$c_{1}, c_{2}$	constants	har line origin se kitni door hai
$A = [a_{1} a_{2} b_{1} b_{2}]$	coefficient matrix	grid jahan har column ek variable own karta hai
	"determinant lo"	row-parallelogram ka signed area
$D$	$det A = a_{1} b_{2} - a_{2} b_{1}$	do lines ke beech spread
$D_{x}, D_{y}$	$A$ jisme ek column $c$ se swap hua	(parent note mein build hota hai)

Prerequisite map

Equipment checklist

Ek equation mein letter

x

kya represent karta hai?

Ek unknown — ek placeholder us number ke liye jo hum discover karna chahte hain.

a x + b y = c

ko linear kyun kaha jaata hai?

Unknowns sirf pehli power mein aate hain, isliye iska graph ek seedhi line hai.

a_{2}

mein subscript ka kya matlab hai?

Ek name tag — "equation 2 se

a

" —

a

times

2

NAHI.

Coefficient kya hota hai?

Woh known number jo ek unknown ko multiply karta hai, jaise

b_{1}

equation 1 mein

y

ko multiply karta hai.

A

ka kaunsa column variable

x

ko "own" karta hai?

Column 1 — isme dono

x

-coefficients

a_{1}, a_{2}

hain.

a c b d

ko kaise compute karte hain?

a d - b c

: main diagonal (down-right) minus anti-diagonal (down-left).

Determinant geometrically kya measure karta hai?

Matrix ki do rows se bane parallelogram ka signed area.

det A = 0

ka do lines ke baare mein kya matlab hai?

Rows parallel hain, isliye lines ek single point par cross nahi karti.

Cramer's rule kyun kaam nahi karta jab

D = 0

hota hai?

Woh zero se divide karta, jo undefined hai — yeh parallel (non-crossing) lines ki geometry se match karta hai.

Kya seedhe bars

ek grid ya ek number dete hain?

Ek single number — determinant.