Cramer's Rule for solving 2x2 system of equations

Cramer's Rule for Solving Linear Systems

1️⃣ Understand the concept of Cramer's Rule and its systematic approach

2️⃣ Learn to construct coefficient matrices from linear systems

3️⃣ Calculate determinants for 2×2 matrices

4️⃣ Apply the substitution method to find variable-specific determinants

5️⃣ Solve complete 2×2 linear systems using Cramer's Rule

① Introduction to Cramer's Rule

Cramer's Rule is a systematic method for solving linear systems that relies on calculating determinants of matrices derived from the system of equations. What makes it special is its organized, algorithmic approach that can be easily programmed into computers.

Cramer's Rule uses determinants to find exact solutions systematically

The rule works by extracting coefficient matrices from our linear system and then calculating specific determinants to find each variable independently. This makes it particularly useful for computer algorithms and automated solving.

Key Advantage: Cramer's Rule provides a structured, step-by-step approach that works consistently for any solvable linear system

② Setting Up the Coefficient Matrix

For a 2×2 linear system, we first extract the coefficient matrix from our equations. Let's work with this example system:

Example System

5x - 6y = 15

3x + 4y = -29

From this system, we extract the coefficient matrix by taking the coefficients of the variables. Pay careful attention to the signs!

Coefficient Matrix =

\begin{pmatrix} 5 & -6 \\ 3 & 4 \end{pmatrix}

③ Calculating the Main Determinant

The first step in Cramer's Rule is calculating the determinant of the coefficient matrix. For a 2×2 matrix, we use the formula we learned in previous lessons.

Calculating the Main Determinant

Step 1: Take the coefficient matrix

\begin{pmatrix} 5 & -6 \\ 3 & 4 \end{pmatrix}

Step 2: Apply the 2×2 determinant formula
Main diagonal product - Secondary diagonal product

Step 3: Calculate

D = (5)(4) - (-6)(3) = 20 + 18 = 38

Main Determinant: D = 38

④ Applying Cramer's Rule Formula

Now we can set up the Cramer's Rule formulas. Each variable equals a specific determinant divided by the main determinant:

Cramer's Rule Formulas

x = \frac{D_x}{D} = \frac{D_x}{38}

y = \frac{D_y}{D} = \frac{D_y}{38}

⑤ Finding D_x (Determinant for x)

To find D_x, we replace the x-column in our coefficient matrix with the constants from the right side of our equations.

Step-by-Step: Finding D_x

Original coefficient matrix:

\begin{pmatrix} 5 & -6 \\ 3 & 4 \end{pmatrix}

Replace the x-column with constants (15, -29):

\begin{pmatrix} 15 & -6 \\ -29 & 4 \end{pmatrix}

Calculate the determinant:

D_x = (15)(4) - (-6)(-29) = 60 - 174 = -114

Therefore:

x = \frac{D_x}{D} = \frac{-114}{38} = -3

⑥ Finding D_y (Determinant for y)

Similarly, to find D_y, we replace the y-column in our coefficient matrix with the constants.

Step-by-Step: Finding D_y

Original coefficient matrix:

\begin{pmatrix} 5 & -6 \\ 3 & 4 \end{pmatrix}

Replace the y-column with constants (15, -29):

\begin{pmatrix} 5 & 15 \\ 3 & -29 \end{pmatrix}

Calculate the determinant:

D_y = (5)(-29) - (15)(3) = -145 - 45 = -190

Therefore:

y = \frac{D_y}{D} = \frac{-190}{38} = -5

⑦ Final Solution and Verification

Let's summarize our complete solution and verify it works in both original equations.

Complete Solution

x = -3, \quad y = -5

Verification:

Equation 1:

5(-3) - 6(-5) = -15 + 30 = 15

✅

Equation 2:

3(-3) + 4(-5) = -9 - 20 = -29

✅

Summary of Cramer's Rule Process:

Extract coefficient matrix from the linear system
Calculate main determinant D
For each variable, replace its column with constants and calculate new determinant
Divide each variable's determinant by main determinant
Verify solution in original equations

Summary of Key Points

1️⃣ Cramer's Rule uses determinants systematically to solve linear systems

2️⃣ Extract coefficient matrix carefully, watching signs

3️⃣ Calculate main determinant D from coefficient matrix

4️⃣ Replace columns with constants to find variable-specific determinants

5️⃣ Each variable = (its determinant) ÷ (main determinant)