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Linear Equations in Two Variables

Introduction

Linear equations in two variables algebra का एक core concept है जहाँ दो unknowns (आमतौर पर x और y) एक first-degree equation के through जुड़े होते हैं।

यह pattern इसलिए महत्वपूर्ण है क्योंकि यह सिखाता है कि दो lines कहाँ intersect करती हैं - यानी वह common solution जो दोनों equations को satisfy करता है। ऐसे सवाल algebra, geometry और real-life situations जैसे cost-profit analysis में बहुत उपयोग होते हैं।

Pattern: Linear Equations in Two Variables

Pattern

Key idea: ऐसे x और y ढूँढना जो दोनों equations को एक साथ satisfy करें।

General form है:
a₁x + b₁y = c₁ और a₂x + b₂y = c₂

Common solving methods:

  • Substitution Method: एक variable को दूसरे के terms में लिखकर substitute करें।
  • Elimination Method: equations को multiply करके add/subtract करें ताकि एक variable eliminate हो जाए।
  • Cross Multiplication Method: जब equations standard form में हों, quick calculation के लिए उपयोगी।

Step-by-Step Example

Question

x और y का मान निकालें:
2x + 3y = 12
3x + 2y = 13

Solution

  1. Step 1: दोनों equations लिखें

    (1) 2x + 3y = 12
    (2) 3x + 2y = 13

  2. Step 2: x के coefficients equal बनाएं

    (1) को 3 से multiply करें → 6x + 9y = 36
    (2) को 2 से multiply करें → 6x + 4y = 26

  3. Step 3: Subtraction से x eliminate करें

    (6x + 9y) - (6x + 4y) = 36 - 26
    ⇒ 5y = 10
    ⇒ y = 2

  4. Step 4: y = 2 को substitute करके x निकालें

    (1) में रखें: 2x + 3(2) = 12
    ⇒ 2x + 6 = 12
    ⇒ 2x = 6
    ⇒ x = 3

  5. Final Answer:

    x = 3, y = 2

  6. Quick Check:

    (2) में रखकर देखें: 3(3) + 2(2) = 9 + 4 = 13 ✅

Quick Variations

1. Substitution Method तब अपनाएँ जब किसी variable का coefficient simple हो।

2. Cross Multiplication direct calculation देता है।

3. Two-condition वाले word problems - जैसे cost और quantity या ages।

Trick to Always Use

  • Step 1: Equations को standard form (ax + by = c) में arrange करें।
  • Step 2: उस variable को चुनें जिसे eliminate करना easiest हो।
  • Step 3: दूसरे variable को substitute करके final answer निकालें।

Summary

Summary

Linear Equations in Two Variables pattern में:

  • Equations को solve करने से पहले properly arrange करें।
  • Elimination तब use करें जब coefficients easily match हो सकें।
  • अपने (x, y) pair को दोनों equations में verify जरूर करें।

Practice

(1/5)
1. Solve the equations: x + y = 10 and x - y = 2
easy
A. x=6, y=4
B. x=5, y=5
C. x=4, y=6
D. x=7, y=3

Solution

  1. Step 1: Add the equations

    (x + y) + (x - y) = 10 + 2 ⇒ 2x = 12.

  2. Step 2: Solve for x

    x = 12 ÷ 2 = 6.

  3. Step 3: Substitute x to find y

    Substitute x = 6 into x + y = 10 ⇒ 6 + y = 10 ⇒ y = 4.

  4. Final Answer:

    x = 6, y = 4 → Option A.

  5. Quick Check:

    6 - 4 = 2 ✔
Hint: Add equations directly when one variable cancels out.
Common Mistakes: Incorrectly adding terms or misplacing signs during substitution.
2. Solve the equations: 2x + y = 11 and x + y = 8
easy
A. x=4, y=4
B. x=3, y=5
C. x=5, y=3
D. x=2, y=6

Solution

  1. Step 1: Subtract the equations

    (2x + y) - (x + y) = 11 - 8 ⇒ x = 3.

  2. Step 2: Substitute x to find y

    3 + y = 8 ⇒ y = 5.

  3. Final Answer:

    x = 3, y = 5 → Option B.

  4. Quick Check:

    2(3) + 5 = 11 ✔
Hint: When y has the same coefficient, subtract to eliminate it instantly.
Common Mistakes: Sign errors when subtracting equations.
3. Solve the equations: 3x + 2y = 12 and 2x + 3y = 13
easy
A. x=2, y=3
B. x=3, y=2
C. x=4, y=1
D. x=1, y=4

Solution

  1. Step 1: Scale both equations to match x-coefficients

    Multiply (1) by 2 → 6x + 4y = 24; multiply (2) by 3 → 6x + 9y = 39.

  2. Step 2: Subtract to eliminate x

    (6x + 9y) - (6x + 4y) = 39 - 24 ⇒ 5y = 15 ⇒ y = 3.

  3. Step 3: Substitute y to find x

    3x + 6 = 12 ⇒ 3x = 6 ⇒ x = 2.

  4. Final Answer:

    x = 2, y = 3 → Option A.

  5. Quick Check:

    2(2) + 3(3) = 4 + 9 = 13 ✔
Hint: Multiply equations to equalize coefficients and eliminate one variable.
Common Mistakes: Forgetting to multiply all terms or subtracting in the wrong order.
4. Solve the equations: 4x + 3y = 18 and 3x + 2y = 13
medium
A. x=2, y=4
B. x=4, y=2
C. x=2, y=3
D. x=3, y=2

Solution

  1. Step 1: Scale equations to match x-coefficients

    (1)×3 → 12x + 9y = 54; (2)×4 → 12x + 8y = 52.

  2. Step 2: Subtract to eliminate x

    (12x + 9y) - (12x + 8y) = 54 - 52 ⇒ y = 2.

  3. Step 3: Substitute y to find x

    3x + 4 = 13 ⇒ 3x = 9 ⇒ x = 3.

  4. Final Answer:

    x = 3, y = 2 → Option D.

  5. Quick Check:

    4(3) + 3(2) = 12 + 6 = 18 ✔
Hint: Use LCM of coefficients to eliminate variables cleanly.
Common Mistakes: Not multiplying each term properly before subtracting.
5. Solve the equations: 5x + 4y = 24 and 3x + 2y = 14
medium
A. x=3, y=2
B. x=2, y=3
C. x=4, y=1
D. x=2, y=4

Solution

  1. Step 1: Scale to match y-coefficients

    Multiply (2) by 2 → 6x + 4y = 28.

  2. Step 2: Subtract equations to find x

    (5x + 4y) - (6x + 4y) = 24 - 28 ⇒ -x = -4 ⇒ x = 4.

  3. Step 3: Substitute x to find y

    3(4) + 2y = 14 ⇒ 12 + 2y = 14 ⇒ y = 1.

  4. Final Answer:

    x = 4, y = 1 → Option C.

  5. Quick Check:

    5(4) + 4(1) = 20 + 4 = 24 ✔
Hint: Match coefficients of one variable, subtract, then substitute.
Common Mistakes: Dropping negative signs during subtraction.

Mock Test

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