SOLUTION: 5. At $1.40 per pound, the daily supply of tobacco is 1075 pounds, and the daily demand is 580 pounds. When the price falls to $1.20 per pound, the daily supply decreases to 575

Question 1189902: 5. At $1.40 per pound, the daily supply of tobacco is 1075 pounds, and the daily
demand is 580 pounds. When the price falls to $1.20 per pound, the daily
supply decreases to 575 pounds and the daily demand increases to 980 pounds.
Assume that the supply and demand equations are linear.
a) Find the supply equations.
b) Find the demand equations.
c) Find the equilibrium price and quantity.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part A

x = price in dollars
y = amount supplied in pounds

"At $1.40 per pound, the daily supply of tobacco is 1075 pounds" gives the point (1.40,1075)

" When the price falls to $1.20 per pound, the daily
supply decreases to 575 pounds " means we have another point (1.20,575)

Slope through those points
m = (y2-y1)/(x2-x1)
m = (575-1075)/(1.20-1.40)
m = (-500)/(-0.20)
m = 2500
The slope of 2500 means that for every price increase of $1, the amount supplied goes up by 2500 units.

Let's find the y intercept using the slope and the first point mentioned
y = mx+b
1075 = 2500*(1.40)+b
1075 = 3500+b
1075-3500 = b
-2425 = b
b = -2425
Admittedly, the y intercept doesn't make much sense. It's not possible to have a negative amount supplied. However, this is useful to help set up the equation y = 2500x-2425. The key thing to keep in mind is that there's a certain domain to stick to. It only makes sense to have x > 0 and y > 0.

To check this, we plug in x = 1.40 and should get y = 1075. Also, plugging in x = 1.20 should lead to y = 575. I'll let you do those checks.

Answer:

The supply equation is
y = 2500x-2425

==================================================

Part B

x = price in dollars
y = amount demanded in pounds

We'll do similar steps for the demand equation
"At $1.40 per pound... the daily
demand is 580 pounds."
We have the point (1.40,580)

Also,
"the price falls to $1.20 per pound...the daily demand increases to 980 pounds."
So we have the other point (1.20, 980)

Slope:
m = (y2-y1)/(x2-x1)
m = (980-580)/(1.20-1.40)
m = 400/(-0.20)
m = -2000
A negative slope tells us that each time the price (x) goes up, the demand (y) goes down.
Specifically, each time the price goes up by $1, the demand goes down by 2000 pounds.
Unlike the negative y intercept not making much sense, the slope can be negative. It's always negative for demand equations. The price and amount demanded move in opposite directions for demand equations.

Like before, I'll use the slope and first point to find the y intercept
y = mx+b
580 = -2000*(1.40)+b
580 = -2800+b
580+2800 = b
b = 3380

The equation
y = mx+b
becomes
y = -2000x+3380

To check this, we plug in x = 1.40 and should get y = 580. Also, plugging in x = 1.20 should lead to y = 980. I'll let you do those checks.

Answer:

The demand equation is
y = -2000x+3380

==================================================

Part C

We found from the previous parts that:
Supply equation is y = 2500x-2425
Demand equation is y = -2000x+3380

Equilibrium is reached when the quantity supplied equals quantity demanded. Visually, it's where the supply and demand lines intersect.

Let's use substitution to find that solution
y = -2000x+3380
2500x-2425 = -2000x+3380
2500x+2000x = 3380+2425
4500x = 5805
x = 5805/4500
x = 1.29

The equilibrium price is $1.29 per pound.

Plug that x value into either y equation
Supply:
y = 2500x-2425
y = 2500*1.29-2425
y = 800
Or,
Demand:
y = -2000x+3380
y = -2000*1.29+3380
y = 800
The equilibrium quantity is 800 pounds.

Answers:
Equilibrium price = $1.29 per pound.
Equilibrium quantity = 800 pounds

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