) Suppose a market research company finds that at a price of p = $30, they would sell x = 42 tiles each month. If they lower the price to p = $20, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,p).
To find the equation of the demand line, we can use the slope-intercept form of a linear equation, which is:
p = mx + b
where p represents the price, x represents the quantity demanded, m represents the slope, and b represents the y-intercept.
We are given two points on the line:
- At p = $30, x = 42
- At p = $20, x = 52
We can use these points to find the values of m and b.
Step 1: Find the slope (m)
The slope of a line can be calculated using the formula:
m = (p2 - p1) / (x2 - x1)
Given the points (42, $30) and (52, $20), we can substitute the values into the formula:
m = (20 - 30) / (52 - 42)
= -10 / 10
= -1
So, the slope of the demand line is -1.
Step 2: Find the y-intercept (b)
To find the y-intercept, we can use one of the given points and substitute the values into the equation:
$30 = (-1)(42) + b
Simplifying the equation:
$30 = -42 + b
Adding 42 to both sides:
$72 = b
So, the y-intercept (b) is $72.
Step 3: Write the equation of the demand line
Using the values of m and b, we can write the equation of the demand line in the form p = mx + b:
p = -x + 72
Therefore, the equation of the demand line is p = -x + 72.