2.1.9
Ek akela equation jaise 2 x + 3 y = 12 infinitely many solutions hote hain — ek straight line par har point. Ek single pair ( x , y ) doosra equation chahiye. Ek system solve karna matlab do lines ka crossing point dhundhna.
KYUN zaroori hai: almost har "do-unknown" wali real problem (coins ka mix, boat aur stream ki speed, do items ki cost) do linear equations ban jaati hai.KYA dhundh rahe hain: intersection point ( x , y ) KAISE : graphically (dono lines draw karo) ya algebraically (substitution / elimination).
Definition Do variables mein linear equationEk equation jis ki form ho
a x + b y + c = 0 , ( a , b ) = ( 0 , 0 ) a , b , c x , y degree 1 ke (koi x 2 x y x straight line hota hai.
KYUN straight line? y y = − b a x − b c y = m x + k m x y
a 1 x + b 1 y a 2 x + b 2 y = c 1 = c 2
Teen geometric possibilities:
KYUN yeh ratios? Line 1 ka slope hai − a 1 / b 1 − a 2 / b 2 a 2 a 1 = b 2 b 1 parallel ya same . Agar constant ratio bhi match kare, toh ek equation doosre ka sirf ek multiple hai → same line . Agar nahi, toh woh alag-alag shifted hain → parallel, no solution .
Intuition Ek equation mein ek variable isolate karo, phir use "plug in" karo — yeh do unknowns ko ek mein badal deta hai.
Intuition Equations ko scale karo taaki ek variable kasame coefficient ho, phir use khatam karne ke liye add/subtract karo. KYUN yeh valid hai: ek true equation ke dono sides mein equal quantities add karna use true hi rakhta hai.
Worked example Elimination — unique solution2 x + 3 y = 13 , 3 x − y = 3
Eqn 2 ko 3 se multiply karo: 9 x − 3 y = 9 Kyun? Taaki 3 y y
Eqn 1 mein add karo: ( 2 x + 9 x ) + ( 3 y − 3 y ) = 13 + 9 ⇒ 11 x = 22 ⇒ x = 2
Eqn 2 mein back-substitute karo: 3 ( 2 ) − y = 3 ⇒ y = 3 Kyun eqn 2? Yeh simpler hai.
Answer ( 2 , 3 ) Check karo: 2 ( 2 ) + 3 ( 3 ) = 13 3 ( 2 ) − 3 = 3
Worked example Substitution — word problemEk boat 30 km downstream 3 h mein aur 30 km upstream 5 h mein jaati hai. Boat speed b s
Downstream speed = b + s = 30/3 = 10 = b − s = 30/5 = 6 Kyun add/subtract? Stream downstream mein help karta hai, upstream mein oppose karta hai.
Pehle se: s = 10 − b b − ( 10 − b ) = 6 ⇒ 2 b = 16 ⇒ b = 8
s = 10 − 8 = 2 Boat 8 km/h, stream 2 km/h.
Worked example "No solution" detect karnax + 2 y = 4 2 x + 4 y = 12 2 1 = 4 2 = 2 1 12 4 = 3 1 = 2 1 a 2 a 1 = b 2 b 1 = c 2 c 1 parallel, no solution . Kyun? Same slope, alag intercepts.
Recall 4 x + 6 y = 20 2 x + 3 y = 10 2 4 = 3 6 = 10 20 = 2 same line, infinitely many solutions . Verify karo: pehle ko 2 se divide karne par exactly doosra milta hai. ✓ Forecast confirmed.
Common mistake "Terms move karne par sign flip hota hai"Galat jo sahi lagta hai: tum jaldi mein likhte ho 3 x − y = 3 ⇒ y = 3 x + 3 Kyun tempting lagta hai: − y Fix: 3 x − y = 3 ⇒ − y = 3 − 3 x ⇒ y = 3 x − 3 Har move hone wale term ka sign flip hota hai.
Common mistake "Parallel lines ka ek solution hota hai"Sahi lagta hai: do lines "zaroor" kahin milti hongi. Fix: parallel lines ka equal slope hota hai aur woh kabhi nahi miltiñ — ratio test a 2 a 1 = b 2 b 1 = c 2 c 1 no solution signal karta hai.
Common mistake "Sirf ek equation check karna"( x , y ) Fix: point ko dono satisfy karna chahiye — guarantee karne ke liye ki yeh intersection hai, dono mein substitute karo.
Recall Ek 12-saal ke bacche ko explain karoSocho ek map par do seedhi sadkein hain. Ek equation = ek sadak. Answer dhundhna matlab exactly woh jagah dhundhna jahan do sadkein cross karti hain — wahi ( x , y )
"SEC" teen cases ke liye
S ame line = sab ratios Same hain. E qual-slope-only = parallel, no solution. C ross once = coefficients differ (unique). Aur method choice ke liye: "Ek variable akela ho? Substitution. Coefficients tidy ho? Elimination."
Do variables mein linear equation ki general form? a x + b y + c = 0 ( a , b ) = ( 0 , 0 )
Do variables mein EK linear equation ke kitne solutions hote hain? Infinitely many (apni line par har point).
UNIQUE solution ke liye ratio condition? a 2 a 1 = b 2 b 1
NO solution (parallel) ke liye ratio condition? a 2 a 1 = b 2 b 1 = c 2 c 1
INFINITE solutions (same line) ke liye ratio condition? a 2 a 1 = b 2 b 1 = c 2 c 1
Elimination (Cramer) se x x = a 1 b 2 − a 2 b 1 c 1 b 2 − c 2 b 1
Denominator D = a 1 b 2 − a 2 b 1 = 0 Lines parallel ya same hain → koi unique solution nahi.
Boat b s Downstream = b + s = b − s
Graph straight line kyun hota hai? y = − b a x − b c x y
Jab ek coefficient 1 Substitution (us variable ko cleanly isolate karo).
Slope-intercept form y = mx + c Simultaneous equations by matrices & determinants Consistency and rank of linear systems Word problems — age, boat-stream, mixture Graphing lines and intercepts Linear inequalities in two variables
Infinitely many solutions
Graphical: find intersection
Cramer formula with D=a1b2-a2b1
