in a certain week a businessman bought 36 bicycles and 32 radios for total of kshs 227280. In the following week he bought 28 bicycles and 24 radios for a total of Kshs 174960. Using matrix method, find the price of each bicycles and each radio that he bought
You have AX = B
(36 32)(x) = (227280)
(28 24)(y)....(174960)
now just solve for X = A-1B
Bicycle 200 radio 250
Bicycle 4500
Radio 2040
To find the price of each bicycle and each radio that the businessman bought, we can set up a system of equations using the matrix method.
Let's denote the price of each bicycle as 'b' and the price of each radio as 'r'.
In the first week, the businessman bought 36 bicycles and 32 radios for a total of Kshs 227,280. We can represent this information as follows:
36b + 32r = 227280
In the following week, he bought 28 bicycles and 24 radios for a total of Kshs 174,960. We can represent this information as:
28b + 24r = 174960
We can write these equations in matrix form as:
⎡36 32⎤ ⎡b⎤ ⎡227280⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣28 24⎦ ⎣r⎦ = ⎣174960⎦
To solve for 'b' and 'r', we can multiply the inverse of the coefficient matrix by the constant matrix:
⎡b⎤ 1/((36*24)-(32*28)) ⎡ 24 -32⎤ ⎡227280⎤
⎢ ⎥ = ⎢ ⎥ ⎢ ⎥
⎣r⎦ ⎣ -28 36⎦ ⎣174960⎦
Now, let's calculate the inverse of the coefficient matrix:
⎡ 24 -32⎤
⎢ ⎥ = 1/((36*24)-(32*28)) * ⎡ 36 -32⎤
⎣ -28 36⎦ ⎣-28 36⎦
= 1/((36*24)-(32*28)) * ⎡ 36 -32⎤
⎣-28 36⎦
= 1/(-256) * ⎡ 36 -32⎤
⎣-28 36⎦
Now, let's multiply the inverse of the coefficient matrix by the constant matrix:
= 1/(-256) * ⎡ 36 -32⎤ ⎡227280⎤
⎣-28 36⎦ ⎣174960⎦
= 1/(-256) * ⎡ (36*227280) + (-32*174960)⎤
⎣ (-28*227280) + (36*174960)⎦
= 1/(-256) * ⎡ 8210880⎤
⎣ 3934080⎦
= ⎡ -8210880/256⎤
⎣ -3934080/256⎦
= ⎡ -32145⎤
⎣ -15345⎦
Therefore, the solution to the system of equations is:
b = -32145
r = -15345
Since negative prices don't make sense in this context, we can disregard the negative signs. Thus, the price of each bicycle is Kshs 32145, and the price of each radio is Kshs 15345.