Sure, let's tackle the problem step by step.
### PART A
#### Copy and complete the table
| | Mrs Ndoro | Mrs Kori | Miss Ndlovu |
|-------------------|-----------|----------|-------------|
| Bananas (kg) | 1.2 | 2.2 | 1.8 |
| Apples (kg) | 0.8 | 2.0 | 1.6 |
| Total Paid (\[tex]$) | 2.40 | 5.20 | 4.2 |
The missing value for the total amount paid by Miss Ndlovu is \(4.2\).
#### Define the variables
Let:
- \( x \) be the price per kilogram for Bananas.
- \( y \) be the price per kilogram for Apples.
#### Form separate equations using the information from each column
Mrs Ndoro's Equation:
\[ 1.2x + 0.8y = 2.40 \]
Mrs Kori's Equation:
\[ 2.2x + 2.0y = 5.20 \]
Miss Ndlovu's Equation:
\[ 1.8x + 1.6y = 4.2 \]
### PART B
#### Solve for the unknown variables using the matrix method
##### Pulling out the matrix
To solve the system of linear equations defined by Mrs Ndoro and Mrs Kori, we can write it in matrix form \( A \mathbf{x} = \mathbf{b} \).
\[ A = \begin{pmatrix}
1.2 & 0.8 \\
2.2 & 2.0
\end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix}
x \\
y
\end{pmatrix} \]
\[ \mathbf{b} = \begin{pmatrix}
2.40 \\
5.20
\end{pmatrix} \]
##### Finding the determinant of the drawn matrix
To find \( x \) and \( y \), we first need to check that the determinant of \( A \) is non-zero.
\[ \text{Det}(A) = \begin{vmatrix}
1.2 & 0.8 \\
2.2 & 2.0
\end{vmatrix} = (1.2 \cdot 2.0) - (0.8 \cdot 2.2) \]
\[ \text{Det}(A) = 2.4 - 1.76 = 0.64 \]
Since the determinant is \( 0.64 \), which is not zero, we can proceed to solve the system using the inverse of \( A \).
#### Solving the system
The inverse of matrix \( A \) is given by
\[ A^{-1} = \frac{1}{\text{Det}(A)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix} \]
where \( a = 1.2 \), \( b = 0.8 \), \( c = 2.2 \), and \( d = 2.0 \).
\[ A^{-1} = \frac{1}{0.64} \begin{pmatrix}
2.0 & -0.8 \\
-2.2 & 1.2
\end{pmatrix} = \begin{pmatrix}
3.125 & -1.25 \\
-3.4375 & 1.875
\end{pmatrix} \]
Now we can find \( \mathbf{x} \) by multiplying \( A^{-1} \) with \( \mathbf{b} \):
\[ \mathbf{x} = A^{-1} \mathbf{b} = \begin{pmatrix}
3.125 & -1.25 \\
-3.4375 & 1.875
\end{pmatrix} \begin{pmatrix}
2.40 \\
5.20
\end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix}
(3.125 \cdot 2.40) + (-1.25 \cdot 5.20) \\
(-3.4375 \cdot 2.40) + (1.875 \cdot 5.20)
\end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix}
7.5 - 6.5 \\
-8.25 + 9.75
\end{pmatrix} \]
\[ \mathbf{x} = \begin{pmatrix}
1.00 \\
1.50
\end{pmatrix} \]
Thus, the price per kilogram for Bananas is \( \$[/tex]1.00 \) and the price per kilogram for Apples is [tex]\( \$1.50 \)[/tex].
Finally, we use these prices to calculate the total amount paid by Miss Ndlovu:
[tex]\[ 1.8x + 1.6y = 1.8 \cdot 1.00 + 1.6 \cdot 1.50 = 1.8 + 2.4 = 4.2 \][/tex]
So, Miss Ndlovu paid [tex]\( \$4.20 \)[/tex] for her fruits.
