find the value of x, y, z with the help of determinant method
4x+3y+2z=60
x+2y+3z=45
6x+2y+3z=70

Answer

To solve the system of equations using the determinant method, we will follow these steps:

### Given Equations:
1. \( 4x + 3y + 2z = 60 \) (Equation 1)
2. \( x + 2y + 3z = 45 \) (Equation 2)
3. \( 6x + 2y + 3z = 70 \) (Equation 3)

### Step 1: Write the Coefficient Matrix and the Constant Matrix
The coefficient matrix \( A \) and the constant matrix \( B \) can be represented as follows:

\[ A = \begin{bmatrix} 4 & 3 & 2 \\ 1 & 2 & 3 \\ 6 & 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 60 \\ 45 \\ 70 \end{bmatrix} \]

### Step 2: Calculate the Determinant of Matrix \( A \)
To find the determinant of matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

\[ \text{det}(A) = 4(2 \cdot 3 - 3 \cdot 2) - 3(1 \cdot 3 - 3 \cdot 6) + 2(1 \cdot 2 - 2 \cdot 6) \]

Calculating each term:

1. \( 4(6 - 6) = 4 \cdot 0 = 0 \)
2. \( -3(3 - 18) = -3 \cdot (-15) = 45 \)
3. \( 2(2 - 12) = 2 \cdot (-10) = -20 \)

Thus,

\[ \text{det}(A) = 0 + 45 - 20 = 25 \]

### Step 3: Calculate Determinants for \( x, y, z \)
Next, we will find the determinants for \( x, y, z \) by replacing the respective columns of \( A \) with the constant matrix \( B \).

#### Determinant for \( x \) (denoted as \( D_x \)):
Replace the first column of \( A \) with \( B \):

\[ D_x = \begin{vmatrix} 60 & 3 & 2 \\ 45 & 2 & 3 \\ 70 & 2 & 3 \end{vmatrix} \]

Calculating \( D_x \):

\[ D_x = 60(2 \cdot 3 - 3 \cdot 2) - 3(45 \cdot 3 - 70 \cdot 2) + 2(45 \cdot 2 - 70 \cdot 2) \]

Calculating each term:

1. \( 60(6 - 6) = 60 \cdot 0 = 0 \)
2. \( -3(135 - 140) = -3 \cdot (-5) = 15 \)
3. \( 2(90 - 140) = 2 \cdot (-50) = -100 \)

Thus,

\[ D_x = 0 + 15 - 100 = -85 \]

#### Determinant for \( y \) (denoted as \( D_y \)):
Replace the second column of \( A \) with \( B \):

\[ D_y = \begin{vmatrix} 4 & 60 & 2 \\ 1 & 45 & 3 \\ 6 & 70 & 3 \end{vmatrix} \]

Calculating \( D_y \):

\[ D_y = 4(45 \cdot 3 - 3 \cdot 70) - 60(1 \cdot 3 - 6 \cdot 2) + 2(1 \cdot 70 - 6 \cdot 45) \]

Calculating each term:

1. \( 4(135 - 210) = 4 \cdot (-75) = -300 \)
2. \( -60(3 - 12) = -60 \cdot (-9) = 540 \)
3. \( 2(70 - 270) = 2 \cdot (-200) = -400 \)

Thus,

\[ D_y = -300 + 540 - 400 = -160 \]

#### Determinant for \( z \) (denoted as \( D_z \)):
Replace the third column of \( A \) with \( B \):

\[ D_z = \begin{vmatrix} 4 & 3 & 60 \\ 1 & 2 & 45 \\ 6 & 2 & 70 \end{vmatrix} \]

Calculating \( D_z \):

\[ D_z = 4(2 \cdot 70 - 45 \cdot 2) - 3(1 \cdot 70 - 6 \cdot 45) + 60(1 \cdot 2 - 6 \cdot 2) \]

Calculating each term:

1. \( 4(140 - 90) = 4 \cdot 50 = 200 \)
2. \( -3(70 - 270) = -3 \cdot (-200) = 600 \)
3. \( 60(2 - 12) = 60 \cdot (-10) = -600 \)

Thus,

\[ D_z = 200 + 600 - 600 = 200 \]

### Step 4: Calculate Values of \( x, y, z \)
Using Cramer's Rule:

\[ x = \frac{D_x}{\text{det}(A)} = \frac{-85}{25} = -3.4 \]
\[ y = \frac{D_y}{\text{det}(A)} = \frac{-160}{25} = -6.4 \]
\[ z = \frac{D_z}{\text{det}(A)} = \frac{200}{25} = 8 \]

### Final Values:
\[ x = -3.4, \quad y = -6.4, \quad z = 8 \]