Comprehension
On her birthday, Prema decides to donate some money to children of an orphanage home.
If there are 8 children less, everyone gets ₹ 10 more. However, if there are 16 children more, everyone gets ₹ 10 less. Let the number of children in the orphanage home be \( x \) and the amount to be donated to each child be \( y \).
Based on the above information, answer the following questions:
Question: 1
Write the system of linear equations in \( x \) and \( y \) formed from the given situation.
Write the system of linear equations in \( x \) and \( y \) formed from the given situation.
Show Hint
To form equations from a word problem:
1. Use the condition that the total remains constant (e.g., total donation).
2. Write separate equations for the given scenarios and simplify them.
Updated On: Feb 11, 2025
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Solution and Explanation
Step 1: Total amount donated is the same in both cases.
- Case 1: When there are \( x - 8 \) children, each gets \( y + 10 \):
\[
(x - 8)(y + 10) = xy.
\]
Simplify:
\[
xy - 8y + 10x - 80 = xy \quad \Rightarrow \quad -8y + 10x = 80 \quad \Rightarrow \quad 10x - 8y = 80. \quad \cdots (1)
\]
- Case 2: When there are \( x + 16 \) children, each gets \( y - 10 \):
\[
(x + 16)(y - 10) = xy.
\]
Simplify:
\[
xy + 16y - 10x - 160 = xy \quad \Rightarrow \quad 16y - 10x = 160. \quad \cdots (2)
\]
Step 2: The system of linear equations is:
\[
10x - 8y = 80, \quad -10x + 16y = 160.
\]
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Question: 2
Write the system of linear equations, obtained in (i) above, in matrix form \( AX = B \).
Write the system of linear equations, obtained in (i) above, in matrix form \( AX = B \).
Show Hint
To convert a system of linear equations into matrix form:
1. Write the coefficients of variables in a matrix \( A \).
2. Write the variables in a column matrix \( X \) and the constants in \( B \).
Updated On: Feb 11, 2025
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Solution and Explanation
The system of linear equations:
\[
10x - 8y = 80, \quad -10x + 16y = 160.
\]
Matrix form \( AX = B \): \[ A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
Thus, \[ AX = B \quad \text{is written as:} \quad \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
Matrix form \( AX = B \): \[ A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
Thus, \[ AX = B \quad \text{is written as:} \quad \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 80 \\ 160 \end{bmatrix}. \]
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Question: 3
Find the inverse of matrix \( A \).
Find the inverse of matrix \( A \).
Show Hint
To find the inverse of a 2x2 matrix:
1. Compute the determinant \( {det}(A) = ad - bc \).
2. Swap \( a \) and \( d \), and change the signs of \( b \) and \( c \) in the adjoint.
3. Divide each element by \( {det}(A) \).
Updated On: Feb 11, 2025
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Solution and Explanation
Matrix \( A \):
\[
A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}.
\]
Step 1: Compute the determinant of \( A \): \[ \det(A) = (10)(16) - (-8)(-10) = 160 - 80 = 80. \]
Step 2: Compute the adjoint of \( A \): \[ \text{Adj}(A) = \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Step 3: Compute the inverse of \( A \): \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{Adj}(A) = \frac{1}{80} \cdot \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Simplify: \[ A^{-1} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix}. \]
Step 1: Compute the determinant of \( A \): \[ \det(A) = (10)(16) - (-8)(-10) = 160 - 80 = 80. \]
Step 2: Compute the adjoint of \( A \): \[ \text{Adj}(A) = \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Step 3: Compute the inverse of \( A \): \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{Adj}(A) = \frac{1}{80} \cdot \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Simplify: \[ A^{-1} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix}. \]
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Question: 4
Determine the values of \( x \) and \( y \).
Determine the values of \( x \) and \( y \).
Show Hint
To solve for variables using \( X = A^{-1}B \):
1. Multiply the inverse matrix \( A^{-1} \) with \( B \).
2. Perform element-wise multiplication and addition.
Updated On: Feb 11, 2025
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Solution and Explanation
Step 1: Use \( X = A^{-1}B \):
\[
X = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix} \begin{bmatrix} 80 \\ 160 \end{bmatrix}.
\]
Step 2: Perform the matrix multiplication: \[ x = 0.2(80) + 0.1(160) = 16 + 16 = 32, \] \[ y = 0.125(80) + 0.125(160) = 10 + 20 = 30. \]
Final Answer: \[ x = 32, \quad y = 30. \]
Step 2: Perform the matrix multiplication: \[ x = 0.2(80) + 0.1(160) = 16 + 16 = 32, \] \[ y = 0.125(80) + 0.125(160) = 10 + 20 = 30. \]
Final Answer: \[ x = 32, \quad y = 30. \]
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