'Unlock the Secrets of Class 12 Math with Cramer's Rule - New Curriculum Revealed!'

Today's video is all about Class 12 Math Exercise 5.2 on Cramer's rule, which is a new addition to the curriculum. As a student myself, I know how tough math can be, but don't worry, I've got you covered. In this video, I'll be sharing my notes on Cramer's rule, breaking down the concepts, and providing step-by-step solutions to the exercise questions. So grab your notebooks and let's dive into this new topic together.

Cramer's Rule is a method for solving systems of linear equations with three variables. It's a powerful tool for finding the values of the variables that satisfy a system of two linear equations. In this explanation, I'll cover the definition, example, formula, and step-by-step instructions for solving a system of linear equations using Cramer's Rule for 2x2 systems.

**Definition:**

Cramer's Rule is a method for solving systems of linear equations by finding the value of each variable in terms of its corresponding determinant. It's named after the Swiss mathematician Gabriel Cramer, who developed the rule in the 18th century.

Cramer's Rule is a method for solving systems of linear equations of the form:

a11x + a12y = b1
a21x + a22y = b2

...where aij are the coefficients of the variables, and bi are the constants.

**Example:**

Let's consider a simple example:

2x + 3y = 7
x - 2y = -3

To solve this system using Cramer's Rule, we need to find the values of x and y that satisfy both equations.

**Formula:**

The formula for Cramer's Rule is as follows:

x = (Dx / D) where Dx is the determinant of the matrix formed by replacing the first column with the constants, and D is the determinant of the original matrix.

y = (Dy / D) where Dy is the determinant of the matrix formed by replacing the second column with the constants, and D is the same determinant as above.

**Step-by-Step Solution:**

1. Write down the system of linear equations:

2x + 3y = 7
x - 2y = -3

2. Create the augmented matrix:

[ 2 3 | 7 ]
[ 1 -2 | -3 ]

3. Compute the determinant of the original matrix (D):

D = | 2 3 |
| 1 -2 |
|---|
= 2(1) - 3(-2) = 4 + 6 = 10

4. Compute the determinants of the matrices formed by replacing each column with the constants:

Dx = | 7 3 |
| -3 -2 |
|---|
= 7(-2) - 3(-3) = -14 + 9 = -5

Dy = | 2 -3 |
| 1 -2 |
|---|
= 2(-2) - 1(-3) = -4 + 3 = -1

5. Calculate x and y using Cramer's Rule:

x = (Dx / D) = (-5 / 10) = -0.5
y = (Dy / D) = (-1 / 10) = -0.1

So, the solution to the system is x = -0.5 and y = -0.1.

***Let's consider a system of three linear equations with three variables: about 3x3

2x + 3y - z = 7
x - 2y + 3z = -3
4x + y + 2z = 11

We can use Cramer's Rule to find the values of x, y, and z.

**Formula:**

The formula for Cramer's Rule is:

x = D1 / D
y = D2 / D
z = D3 / D

where:

* D is the determinant of the coefficient matrix (the original matrix with the variables)
* D1, D2, and D3 are the determinants of the matrices obtained by replacing the first, second, and third columns of the coefficient matrix with the corresponding column of constants (right-hand side values)

**Step-by-Step Instructions:**

1. Write down the system of linear equations:
2x + 3y - z = 7
x - 2y + 3z = -3
4x + y + 2z = 11

2. Identify the coefficient matrix (the original matrix with the variables):
| 2 3 -1 |
| 1 -2 3 |
| 4 1 2 |

3. Calculate the determinant of the coefficient matrix (D):

D = | 2*1*2 - (3*1*(-1)) - ((-1)*(-2)*4) | = | 4 - (-3) + 8 | = | 7 |

4. Calculate the determinants D1, D2, and D3:
* Replace the first column of the coefficient matrix with the column of constants (right-hand side values):
| 7 -3 -1 |
|-3 -3 3 |
|11 1 2 |
* Calculate the determinant:
D1 = | 7 -3 -1 | = | -(7-15) | = | 8 |
* Replace the second column of the coefficient matrix with the column of constants (right-hand side values):
| 7 7 -1 |
|-3 -6 3 |
|11 0 2 |
* Calculate the determinant:
D2 = | 7 7 -1 | = | -(7-14) | = |-7 |
* Replace the third column of the coefficient matrix with the column of constants (right-hand side values):
| 7 -3 -6 |
|-3 -6 -6 |
|11 -1 -6 |
* Calculate the determinant:
D3 = | 7 -3 -6 | = |-14-(-6)| = |-8 |

5. Use Cramer's Rule to find the values of x, y, and z:
x = D1 / D = 8 / 7 ≈ 1.143
y = D2 / D = (-7) / 7 ≈ -1
z = D3 / D = (-8) / 7 ≈ -1.143

Therefore, the values of x, y, and z are approximately:

x ≈ 1.143
y ≈ -1
z ≈ -1.143

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