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Answer :
To find the equation of demand, we need to determine the relationship between price and quantity based on the given data. We have two points representing two scenarios:
1. At a price of \[tex]$74, 240 pairs of shoes are offered, but only 114 people are interested. Since we're focusing on demand, we consider the quantity offered, which is 240. This gives us our first point: (240, 74).
2. At a price of \$[/tex]50, 210 people are interested in buying the shoes. This gives us our second point: (210, 50).
These points represent the quantity (x) and price (y).
Step-by-Step Solution:
1. Find the Slope (m):
The formula to find the slope [tex]\( m \)[/tex] of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points (240, 74) and (210, 50):
[tex]\[
m = \frac{50 - 74}{210 - 240} = \frac{-24}{-30} = 0.8
\][/tex]
2. Find the Y-Intercept (b):
Using the equation of a line in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex] or simply solving for [tex]\( b \)[/tex] in [tex]\( y = mx + b \)[/tex]:
[tex]\[
b = y_1 - m \cdot x_1
\][/tex]
Substitute one of the points (let's use (240, 74)) and the slope:
[tex]\[
b = 74 - 0.8 \times 240 = 74 - 192 = -118
\][/tex]
3. Write the Equation of Demand:
Now that we have [tex]\( m = 0.8 \)[/tex] and [tex]\( b = -118 \)[/tex], the equation of demand is:
[tex]\[
y = 0.8x - 118
\][/tex]
This equation models the relationship between the quantity of shoes and the price, showing how changes in quantity affect the price, based on the given demand scenarios.
1. At a price of \[tex]$74, 240 pairs of shoes are offered, but only 114 people are interested. Since we're focusing on demand, we consider the quantity offered, which is 240. This gives us our first point: (240, 74).
2. At a price of \$[/tex]50, 210 people are interested in buying the shoes. This gives us our second point: (210, 50).
These points represent the quantity (x) and price (y).
Step-by-Step Solution:
1. Find the Slope (m):
The formula to find the slope [tex]\( m \)[/tex] of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points (240, 74) and (210, 50):
[tex]\[
m = \frac{50 - 74}{210 - 240} = \frac{-24}{-30} = 0.8
\][/tex]
2. Find the Y-Intercept (b):
Using the equation of a line in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex] or simply solving for [tex]\( b \)[/tex] in [tex]\( y = mx + b \)[/tex]:
[tex]\[
b = y_1 - m \cdot x_1
\][/tex]
Substitute one of the points (let's use (240, 74)) and the slope:
[tex]\[
b = 74 - 0.8 \times 240 = 74 - 192 = -118
\][/tex]
3. Write the Equation of Demand:
Now that we have [tex]\( m = 0.8 \)[/tex] and [tex]\( b = -118 \)[/tex], the equation of demand is:
[tex]\[
y = 0.8x - 118
\][/tex]
This equation models the relationship between the quantity of shoes and the price, showing how changes in quantity affect the price, based on the given demand scenarios.
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