Solution of Simultaneous Linear Equations by Cramer’s Rule [ English ]

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1. Introduction

In discrete mathematics and linear algebra, systems of simultaneous linear equations can be solved using several methods. One important and systematic method is Cramer’s Rule.

Cramer’s Rule uses determinants to find the values of unknown variables. It is especially useful for small systems (2×2 and 3×3).

2. Condition for Applying Cramer’s Rule

Cramer’s Rule is applicable only when:

det(A) ≠ 0

✔ The coefficient matrix must be non-singular ✔ The system must have a unique solution

3. System of Two Equations (2×2)

Consider:

a1x + b1y = c1
a2x + b2y = c2

Step 1: Determinant of Coefficient Matrix

D = | a1  b1 |
    | a2  b2 |

Step 2: Determinants for Variables

Dx = | c1  b1 |
     | c2  b2 |

Dy = | a1  c1 |
     | a2  c2 |

Step 3: Formula

x = Dx / D
y = Dy / D

4. Example (2×2 System)

Solve:

2x + 3y = 8
x + 2y = 5

Step 1: Compute D

D = | 2  3 |
    | 1  2 | = (2×2 − 3×1) = 4 − 3 = 1

Step 2: Compute Dx

Dx = | 8  3 |
     | 5  2 | = (8×2 − 3×5) = 16 − 15 = 1

Step 3: Compute Dy

Dy = | 2  8 |
     | 1  5 | = (2×5 − 8×1) = 10 − 8 = 2

Step 4: Final Solution

x = 1/1 = 1
y = 2/1 = 2

5. System of Three Equations (3×3)

Consider:

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

Step 1: Determinant

D = determinant of coefficient matrix

Step 2: Replace Columns

• Replace x-column → Dx
• Replace y-column → Dy
• Replace z-column → Dz

Step 3: Formula

x = Dx / D
y = Dy / D
z = Dz / D

6. Key Observations

• If D = 0:

  • No solution OR infinite solutions

• If D ≠ 0:

  • Unique solution exists

7. Advantages of Cramer’s Rule

• Systematic and formula-based
• Easy for small systems
• Uses determinants directly

8. Limitations

• Not efficient for large systems
• Requires determinant calculation

9. Summary

• Cramer’s Rule solves linear equations using determinants

• For 2 variables:

  • x = Dx / D, y = Dy / D

• Applicable only when:

  • det(A) ≠ 0

• Best suited for 2×2 and 3×3 systems