A linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.
First of all, we would determine a linear function for both supply and demand by finding the slope using this formula as follows:
[tex]Slope,\;m = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope,\;m = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]
Note: price (p) in dollars would be the output and quantity (q) would be the input.
For supply, the points include the following:
Slope, m = (330 - 180)/(2640 - 1440)
Slope, m = 150/1200
Slope, m = 1/8.
Next, we would find the y-intercept at (1,440, 180):
p = qm + c
180 = 1/8(1440) + c
180 = 180 + c
c = 0.
Therefore, the supply equation is p = q/8.
For demand, the points include the following:
Slope, m = (330 - 180)/(1,470 - 2,520)
Slope, m = 150/-1,050
Slope, m = -1/7.
Next, we would find the y-intercept at (2,520, 180):
p = qm + c
180 = -1/7(2,520) + c
180 = -360 + c
c = 180 + 360.
y-intercept, c = 540.
Therefore, the demand equation is p = -q/7 + 540.
In order to find the equilibrium price and quantity, we would set the supply equal to the demand:
q/8 = -q/7 + 540
Multiplying all through by 56, we have:
7q = -8q + 30,240
15q = 30,240
q = 30,240/15
Equilibrium quantity, q = 2,016 items.
For the equilibrium price, we have:
p = q/8
p = 2,016/8
Equilibrium price, p = $252.
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