15. 3 bags and 4 pens together cost 257 rupees whereas 4 bags and 3 pens together cost 324 rupees. Find the cost of a bag and 10 pens. Use Matrix Inversion method to solve the system of equations
Answer (2)
Define variables: Let x be the cost of one bag and y be the cost of one pen.
Formulate equations: 3 x + 4 y = 257 and 4 x + 3 y = 324 .
Solve the system using matrix inversion: Find A − 1 and calculate X = A − 1 B to get x = 75 and y = 8 .
Calculate the cost of 1 bag and 10 pens: 75 + 10 ( 8 ) = 155 . The cost is 155 .
Explanation
Problem Analysis Let's analyze the problem. We are given a system of two linear equations with two variables, the cost of a bag and the cost of a pen. We are asked to find the cost of one bag and ten pens using the matrix inversion method.
Define Variables Let's define our variables: Let x be the cost of one bag in rupees. Let y be the cost of one pen in rupees.
Formulate Equations Now, we can formulate the system of linear equations based on the given information:
3 x + 4 y = 257 4 x + 3 y = 324
Matrix Representation We can represent this system of equations in matrix form as A X = B , where:
A = [ 3 4 4 3 ] , X = [ x y ] , and B = [ 257 324 ]
Calculate Determinant Next, we calculate the determinant of matrix A :
det ( A ) = ( 3 × 3 ) − ( 4 × 4 ) = 9 − 16 = − 7
Find Inverse Matrix Now, we find the inverse of matrix A . The adjugate of A is:
adj ( A ) = [ 3 − 4 − 4 3 ]
Therefore, the inverse of A is:
A − 1 = det ( A ) 1 × adj ( A ) = − 7 1 [ 3 − 4 − 4 3 ] = [ − 7 3 7 4 7 4 − 7 3 ]
Solve for x and y To find the solution matrix X , we multiply the inverse of A with matrix B :
X = A − 1 B = [ − 7 3 7 4 7 4 − 7 3 ] [ 257 324 ]
[ x y ] = [ − 7 3 ( 257 ) + 7 4 ( 324 ) 7 4 ( 257 ) − 7 3 ( 324 ) ] = [ − 7 771 + 7 1296 7 1028 − 7 972 ] = [ 7 525 7 56 ] = [ 75 8 ]
Interpret Results So, x = 75 and y = 8 . This means the cost of one bag is 75 rupees and the cost of one pen is 8 rupees.
Calculate Final Cost Finally, we calculate the cost of 1 bag and 10 pens:
Cost = x + 10 y = 75 + 10 ( 8 ) = 75 + 80 = 155
Final Answer Therefore, the cost of 1 bag and 10 pens is 155 rupees.
Examples
Understanding systems of equations can help in various real-life scenarios, such as resource allocation. For instance, a school administrator might use a system of equations to determine how many teachers to hire and how many textbooks to purchase given a fixed budget. Similarly, a farmer could use such systems to optimize the mix of crops to plant based on land availability and market prices. These problems illustrate how mathematical tools can provide valuable insights for efficient decision-making.
Using matrix inversion, we find that the cost of one bag is 75 rupees and the cost of one pen is 8 rupees. Therefore, the total cost for one bag and ten pens is 155 rupees. This is calculated as 75 + 10 ( 8 ) = 155 .
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