Cramer's rule

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

\begin{cases} a_{11}x + a_{12}y = b_1 \\ a_{21}x + a_{22}y = b_2 \end{cases}

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:

\begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}

After arrangement:

\begin{array}{c|c|c} x & y & = \\ \hline 3 & 2 & 7 \\ 1 & -1 & 1 \end{array}

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

D = \begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix}

Variable coefficients

X Matrix

D_x = \begin{vmatrix} 7 & 2 \\ 1 & -1 \end{vmatrix}

Replace x column

Y Matrix

D_y = \begin{vmatrix} 3 & 7 \\ 1 & 1 \end{vmatrix}

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

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

Calculate Main Determinant (D):

D = \begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix} = 3(-1) - 2(1) = -3 - 2 = -5

Calculate X Determinant (D_x):

D_x = \begin{vmatrix} 7 & 2 \\ 1 & -1 \end{vmatrix} = 7(-1) - 2(1) = -7 - 2 = -9

Calculate Y Determinant (D_y):

D_y = \begin{vmatrix} 3 & 7 \\ 1 & 1 \end{vmatrix} = 3(1) - 7(1) = 3 - 7 = -4

✅ Step 4: Apply Cramer's Rule

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

X Formula

x = \frac{D_x}{D}

Y Formula

y = \frac{D_y}{D}

x = \frac{D_x}{D} = \frac{-9}{-5} = \frac{9}{5} = 1.8

y = \frac{D_y}{D} = \frac{-4}{-5} = \frac{4}{5} = 0.8

🔍 Verify the Solution

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

Verification

First equation:

3x + 2y = 7

3(1.8) + 2(0.8) = 5.4 + 1.6 = 7

✓

Second equation:

x - y = 1

1.8 - 0.8 = 1

✓

Solution is correct: x = 1.8, y = 0.8

🧠 Summary of Steps

Cramer's Rule Steps

Arrange equations - Align like variables under each other
Form main matrix - From variable coefficients
Form variable matrices - By replacing columns
Calculate determinants - Use 2×2 formula
Apply the rule - x = D_x/D, y = D_y/D
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