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