To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the demand function [tex]\(P = a - bQ\)[/tex], given the price and quantity pairs, we will set up and solve two equations based on the information provided.
Given:
1. When the price [tex]\(P\)[/tex] is Rs. 40, the quantity demanded [tex]\(Q\)[/tex] is 60.
2. When the price [tex]\(P\)[/tex] is Rs. 60, the quantity demanded [tex]\(Q\)[/tex] is 57.
We can write these conditions in the form of the demand function equations:
1. [tex]\(40 = a - b \cdot 60\)[/tex]
2. [tex]\(60 = a - b \cdot 57\)[/tex]
These are two linear equations with two unknowns [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
Let's solve these step-by-step:
### Step 1: Write the equations
1. [tex]\(40 = a - 60b\)[/tex]
2. [tex]\(60 = a - 57b\)[/tex]
### Step 2: Subtract the first equation from the second to eliminate [tex]\(a\)[/tex]
Subtracting the first equation from the second:
[tex]\[ (60 = a - 57b) - (40 = a - 60b) \][/tex]
Results in:
[tex]\[ 60 - 40 = (a - 57b) - (a - 60b) \][/tex]
[tex]\[ 20 = a - 57b - a + 60b \][/tex]
[tex]\[ 20 = 3b \][/tex]
### Step 3: Solve for [tex]\(b\)[/tex]
[tex]\[ b = \frac{20}{3} \][/tex]
[tex]\[ b = 6.6667 \][/tex]
### Step 4: Substitute the value of [tex]\(b\)[/tex] back into one of the original equations to find [tex]\(a\)[/tex]
Using the first equation:
[tex]\[ 40 = a - 60b \][/tex]
[tex]\[ 40 = a - 60 \cdot 6.6667 \][/tex]
[tex]\[ 40 = a - 400 \][/tex]
[tex]\[ a = 440 \][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the demand function [tex]\(P = a - bQ\)[/tex] are:
[tex]\[ a = 440 \][/tex]
[tex]\[ b = 6.6667 \][/tex]
So, the demand function is:
[tex]\[ P = 440 - 6.6667Q \][/tex]
This represents the linear relationship between the price [tex]\(P\)[/tex] and the quantity [tex]\(Q\)[/tex] for the given demand conditions.
