Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
### 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.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
