Three chairs and two tables costs Rs. 1850. Five chairs and three tables costs Rs. 2850. Find the cost of four chairs and one table by using matrices.

Answer

To solve the problem of finding the cost of four chairs and one table using matrices, we can follow these steps:

### Step 1: Set up the equations
Let the cost of one chair be Rs. \( x \) and the cost of one table be Rs. \( y \). We can set up the following equations based on the information provided:

1. For three chairs and two tables costing Rs. 1850:
\[ 3x + 2y = 1850 \quad \text{(Equation 1)} \]

2. For five chairs and three tables costing Rs. 2850:
\[ 5x + 3y = 2850 \quad \text{(Equation 2)} \]

### Step 2: Represent the equations in matrix form
We can represent the above equations in matrix form as \( Ax = b \), where:
\[ A = \begin{pmatrix} 3 & 2 \\ 5 & 3 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad b = \begin{pmatrix} 1850 \\ 2850 \end{pmatrix} \]

### Step 3: Find the inverse of matrix \( A \)
To find \( x \), we need to compute the inverse of matrix \( A \). The formula for the inverse of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is:
\[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Calculating the determinant of \( A \):
\[ \text{det}(A) = (3)(3) - (5)(2) = 9 - 10 = -1 \]

Now, we can find the adjoint of \( A \):
\[ \text{adj}(A) = \begin{pmatrix} 3 & -2 \\ -5 & 3 \end{pmatrix} \]

Thus, the inverse of \( A \) is:
\[ A^{-1} = \frac{1}{-1} \begin{pmatrix} 3 & -2 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} -3 & 2 \\ 5 & -3 \end{pmatrix} \]

### Step 4: Multiply \( A^{-1} \) by \( b \)
Now we can find \( x \) by multiplying \( A^{-1} \) with \( b \):
\[ x = A^{-1}b = \begin{pmatrix} -3 & 2 \\ 5 & -3 \end{pmatrix} \begin{pmatrix} 1850 \\ 2850 \end{pmatrix} \]

Calculating the elements:
1. For \( x \):
\[ x = -3(1850) + 2(2850) = -5550 + 5700 = 150 \]

2. For \( y \):
\[ y = 5(1850) - 3(2850) = 9250 - 8550 = 700 \]

Thus, we have:
\[ x = 150, \quad y = 700 \]

### Step 5: Find the cost of four chairs and one table
Now we can find the cost of four chairs and one table:
\[ \text{Cost} = 4x + y = 4(150) + 700 = 600 + 700 = 1300 \]

### Final Answer
The cost of four chairs and one table is Rs. **1300**.

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