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Linear Equations in Two Variables

Introduction

Linear equations in two variables form a core concept in algebra where two unknowns (usually x and y) are related through an equation of the first degree.

This pattern is essential because it teaches how to find the point where two lines intersect - the common solution that satisfies both equations. Such problems are widely used in algebra, geometry, and real-life situations like cost and profit analysis.

Pattern: Linear Equations in Two Variables

Pattern

The key idea: Find values of x and y that satisfy both equations simultaneously.

The general form is:
a₁x + b₁y = c₁ and a₂x + b₂y = c₂

Common solving methods include:

  • Substitution Method: Express one variable in terms of the other, then substitute.
  • Elimination Method: Multiply and add/subtract equations to eliminate one variable.
  • Cross Multiplication Method: Used when equations are in standard form for quick calculation.

Step-by-Step Example

Question

Solve for x and y:
2x + 3y = 12
3x + 2y = 13

Solution

  1. Step 1: Write both equations

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

  2. Step 2: Make coefficients of x equal

    Multiply (1) by 3 → 6x + 9y = 36
    Multiply (2) by 2 → 6x + 4y = 26

  3. Step 3: Eliminate x by subtraction

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

  4. Step 4: Substitute y = 2 to find x

    Substitute in (1): 2x + 3(2) = 12
    ⇒ 2x + 6 = 12
    ⇒ 2x = 6
    ⇒ x = 3

  5. Final Answer:

    x = 3, y = 2

  6. Quick Check:

    Substitute into (2): 3(3) + 2(2) = 9 + 4 = 13 ✅

Quick Variations

1. Using Substitution Method when one variable has a simple coefficient.

2. Using Cross Multiplication for direct computation.

3. Word problems involving two conditions - like cost and quantity or ages.

Trick to Always Use

  • Step 1: Align equations in standard form (ax + by = c).
  • Step 2: Choose the variable with easiest coefficients to eliminate.
  • Step 3: Substitute back to find the second variable.

Summary

Summary

In the Linear Equations in Two Variables pattern:

  • Always arrange equations properly before solving.
  • Use elimination when coefficients can be easily matched.
  • Always verify your (x, y) pair in both equations for correctness.

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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