Cramer's rule

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Cramer's Rule - Solving Linear Systems

1️⃣ Understand Cramer's Rule for solving linear equation systems
2️⃣ Learn how to form matrices and calculate determinants
3️⃣ Master the steps for computing 2×2 determinants
4️⃣ Apply the formula to find values of x and y
5️⃣ Solve practical examples step by step

📐 Introduction to Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. This technique provides an alternative to other methods like substitution, elimination, or matrix inverse methods.

General System of Equations

A system of two linear equations with two unknowns x and y

🔄 Step 1: Arrange the Equations

The first step is to arrange the equations so that the x variables are aligned under each other, and the y variables are aligned under each other.

Example for Illustration

Original equations:

After arrangement:

Now we can easily form the matrices

🏗️ Step 2: Form the Matrices

We form three matrices: the main matrix, the x-matrix, and the y-matrix.

The Three Matrices

Main Matrix

Variable coefficients

X Matrix

Replace x column

Y Matrix

Replace y column

🧮 Step 3: Calculate the Determinants

We calculate the determinant of each matrix using the 2×2 determinant formula.

Calculating Determinants

2×2 Determinant Formula

Calculate Main Determinant (D):

Calculate X Determinant (D_x):

Calculate Y Determinant (D_y):

✅ Step 4: Apply Cramer's Rule

Now we apply Cramer's Rule to find the values of x and y.

Cramer's Rule

X Formula

Y Formula

🔍 Verify the Solution

Let's verify our solution by substituting the values back into the original equations.

Verification

First equation:

✓

Second equation:

✓
Solution is correct: x = 1.8, y = 0.8

🧠 Summary of Steps

Cramer's Rule Steps

  1. Arrange equations - Align like variables under each other
  2. Form main matrix - From variable coefficients
  3. Form variable matrices - By replacing columns
  4. Calculate determinants - Use 2×2 formula
  5. Apply the rule - x = D_x/D, y = D_y/D
  6. Verify solution - Substitute back into original equations

🎯 Important Rules

  • Solution condition: Main determinant D ≠ 0 (otherwise no unique solution exists)
  • Equation arrangement: Essential before forming matrices
  • Column replacement: To form variable matrices, replace the appropriate column
  • Determinant rule: For 2×2 matrix is ad - bc
  • Verification is essential: Always check your solution
283. Relations and Functions
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