Answer :
Alright! Let's solve the given system of equations step-by-step using determinants.
We have two equations based on the given information:
1. [tex]\( 6x + 4y = 200 \)[/tex] (Equation 1)
2. [tex]\( x + y = 42 \)[/tex] (Equation 2)
where:
- [tex]\( x \)[/tex] represents the price per kg of potatoes.
- [tex]\( y \)[/tex] represents the price per kg of tomatoes.
We will solve these equations using determinants, following Cramer's rule.
### Step 1: Write the system of equations in matrix form
The system of equations can be written in matrix form as:
[tex]\[ \begin{bmatrix} 6 & 4 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 200 \\ 42 \end{bmatrix} \][/tex]
### Step 2: Find the determinant of the coefficient matrix
The coefficient matrix is:
[tex]\[ A = \begin{bmatrix} 6 & 4 \\ 1 & 1 \end{bmatrix} \][/tex]
The determinant of [tex]\( A \)[/tex] ([tex]\( \text{det}(A) \)[/tex]) is calculated as:
[tex]\[ \text{det}(A) = (6 \cdot 1) - (4 \cdot 1) = 6 - 4 = 2 \][/tex]
### Step 3: Find the determinants of matrices [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex]
To find the determinants of [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex], we replace the appropriate columns of [tex]\( A \)[/tex] with the constants from the right-hand side of the equations.
#### Matrix [tex]\( A_x \)[/tex]
[tex]\[ A_x = \begin{bmatrix} 200 & 4 \\ 42 & 1 \end{bmatrix} \][/tex]
The determinant of [tex]\( A_x \)[/tex] ([tex]\( \text{det}(A_x) \)[/tex]) is:
[tex]\[ \text{det}(A_x) = (200 \cdot 1) - (4 \cdot 42) = 200 - 168 = 32 \][/tex]
#### Matrix [tex]\( A_y \)[/tex]
[tex]\[ A_y = \begin{bmatrix} 6 & 200 \\ 1 & 42 \end{bmatrix} \][/tex]
The determinant of [tex]\( A_y \)[/tex] ([tex]\( \text{det}(A_y) \)[/tex]) is:
[tex]\[ \text{det}(A_y) = (6 \cdot 42) - (200 \cdot 1) = 252 - 200 = 52 \][/tex]
### Step 4: Apply Cramer's Rule
Using Cramer's Rule, we can find [tex]\( x \)[/tex] and [tex]\( y \)[/tex] by:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{32}{2} = 16 \][/tex]
[tex]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{52}{2} = 26 \][/tex]
Thus, the price per kg of potatoes [tex]\( (x) \)[/tex] is Rs. 16, and the price per kg of tomatoes [tex]\( (y) \)[/tex] is Rs. 26.
### Final Answer
- The price of 1 kg of potatoes is Rs. 16.
- The price of 1 kg of tomatoes is Rs. 26.
We have two equations based on the given information:
1. [tex]\( 6x + 4y = 200 \)[/tex] (Equation 1)
2. [tex]\( x + y = 42 \)[/tex] (Equation 2)
where:
- [tex]\( x \)[/tex] represents the price per kg of potatoes.
- [tex]\( y \)[/tex] represents the price per kg of tomatoes.
We will solve these equations using determinants, following Cramer's rule.
### Step 1: Write the system of equations in matrix form
The system of equations can be written in matrix form as:
[tex]\[ \begin{bmatrix} 6 & 4 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 200 \\ 42 \end{bmatrix} \][/tex]
### Step 2: Find the determinant of the coefficient matrix
The coefficient matrix is:
[tex]\[ A = \begin{bmatrix} 6 & 4 \\ 1 & 1 \end{bmatrix} \][/tex]
The determinant of [tex]\( A \)[/tex] ([tex]\( \text{det}(A) \)[/tex]) is calculated as:
[tex]\[ \text{det}(A) = (6 \cdot 1) - (4 \cdot 1) = 6 - 4 = 2 \][/tex]
### Step 3: Find the determinants of matrices [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex]
To find the determinants of [tex]\( A_x \)[/tex] and [tex]\( A_y \)[/tex], we replace the appropriate columns of [tex]\( A \)[/tex] with the constants from the right-hand side of the equations.
#### Matrix [tex]\( A_x \)[/tex]
[tex]\[ A_x = \begin{bmatrix} 200 & 4 \\ 42 & 1 \end{bmatrix} \][/tex]
The determinant of [tex]\( A_x \)[/tex] ([tex]\( \text{det}(A_x) \)[/tex]) is:
[tex]\[ \text{det}(A_x) = (200 \cdot 1) - (4 \cdot 42) = 200 - 168 = 32 \][/tex]
#### Matrix [tex]\( A_y \)[/tex]
[tex]\[ A_y = \begin{bmatrix} 6 & 200 \\ 1 & 42 \end{bmatrix} \][/tex]
The determinant of [tex]\( A_y \)[/tex] ([tex]\( \text{det}(A_y) \)[/tex]) is:
[tex]\[ \text{det}(A_y) = (6 \cdot 42) - (200 \cdot 1) = 252 - 200 = 52 \][/tex]
### Step 4: Apply Cramer's Rule
Using Cramer's Rule, we can find [tex]\( x \)[/tex] and [tex]\( y \)[/tex] by:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{32}{2} = 16 \][/tex]
[tex]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{52}{2} = 26 \][/tex]
Thus, the price per kg of potatoes [tex]\( (x) \)[/tex] is Rs. 16, and the price per kg of tomatoes [tex]\( (y) \)[/tex] is Rs. 26.
### Final Answer
- The price of 1 kg of potatoes is Rs. 16.
- The price of 1 kg of tomatoes is Rs. 26.
