Cramer's rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns. Let’s denote:
- [tex]\( x \)[/tex] as the cost per kg of sugar,
- [tex]\( y \)[/tex] as the cost per kg of tea.
We are given the following two linear equations:
1. [tex]\( 17x + 4y = 1110 \)[/tex]
2. [tex]\( 8x + 2y = 540 \)[/tex]
We need to determine [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using Cramer's rule.
### Step 1: Form the coefficient matrix and the constants matrix.
The coefficient matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{pmatrix}
17 & 4 \\
8 & 2
\end{pmatrix}
\][/tex]
The constants matrix [tex]\( B \)[/tex] is:
[tex]\[
B = \begin{pmatrix}
1110 \\
540
\end{pmatrix}
\][/tex]
### Step 2: Calculate the determinant of the coefficient matrix [tex]\( A \)[/tex].
[tex]\[
\text{det}(A) = \begin{vmatrix}
17 & 4 \\
8 & 2
\end{vmatrix}
= (17 \cdot 2) - (4 \cdot 8)
= 34 - 32
= 2
\][/tex]
### Step 3: Form matrices [tex]\( A1 \)[/tex] and [tex]\( A2 \)[/tex] to calculate the determinants for Cramer's rule.
Matrix [tex]\( A1 \)[/tex] is formed by replacing the first column of [tex]\( A \)[/tex] with the constants from [tex]\( B \)[/tex]:
[tex]\[
A1 = \begin{pmatrix}
1110 & 4 \\
540 & 2
\end{pmatrix}
\][/tex]
Matrix [tex]\( A2 \)[/tex] is formed by replacing the second column of [tex]\( A \)[/tex] with the constants from [tex]\( B \)[/tex]:
[tex]\[
A2 = \begin{pmatrix}
17 & 1110 \\
8 & 540
\end{pmatrix}
\][/tex]
### Step 4: Calculate the determinants of [tex]\( A1 \)[/tex] and [tex]\( A2 \)[/tex].
[tex]\[
\text{det}(A1) = \begin{vmatrix}
1110 & 4 \\
540 & 2
\end{vmatrix}
= (1110 \cdot 2) - (4 \cdot 540)
= 2220 - 2160
= 60
\][/tex]
[tex]\[
\text{det}(A2) = \begin{vmatrix}
17 & 1110 \\
8 & 540
\end{vmatrix}
= (17 \cdot 540) - (1110 \cdot 8)
= 9180 - 8880
= 300
\][/tex]
### Step 5: Apply Cramer's rule to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
[tex]\[
x = \frac{\text{det}(A1)}{\text{det}(A)}
= \frac{60}{2}
= 30
\][/tex]
[tex]\[
y = \frac{\text{det}(A2)}{\text{det}(A)}
= \frac{300}{2}
= 150
\][/tex]
### Conclusion:
The cost per kg of sugar is Rs. 30 and the cost per kg of tea is Rs. 150.
