For the pair of supply-and-demand equations, where x represents the quantity demanded in units of 1,000 and p is the unit price in dollars, find the equilibrium quantity and the equilibrium price.
2x + 9p - 93 = 0 and 3x - 14p + 108 = 0
a. equilibrium quantity 12,000 units; equilibrium price $9
b. equilibrium quantity 12,000 units; equilibrium price $13
c. equilibrium quantity 6,000 units; equilibrium price $13
d. equilibrium quantity 6,000 units; equilibrium price $9
thank you
Well, well, well, it seems we've got ourselves a classic supply and demand problem! Let's solve it with a touch of humor, shall we?
First, we need to find the equilibrium quantity and price. So, let's set these two equations equal to each other and solve for x and p.
2x + 9p - 93 = 3x - 14p + 108
Hmm, equations battling it out! Looks like we have a rumble in the math jungle!
Rearranging the terms, we get:
3x - 2x = 14p + 9p + 108 + 93
Simplifying, we have:
x = 23p + 201
Now, let's substitute this expression for x into one of the original equations. I choose the first one, just for kicks:
2(23p + 201) + 9p - 93 = 0
Expanding the equation, we get:
46p + 402 + 9p - 93 = 0
Combining like terms:
55p + 309 = 0
Finally, solving for p:
p = -309/55
Oh no, we can't have a negative price, that's no fun!
So, it looks like there's no equilibrium price and the Joker has struck again!
But wait, we still have to find the equilibrium quantity, right? Let's substitute our value of p back into our earlier expression for x:
x = 23p + 201
x = 23(-309/55) + 201
Calculating it out, we get:
x = -1353.09
Oh, dear. Negative quantity? That's laughably unrealistic!
So, based on these calculations, it seems that there is no equilibrium price and quantity for this pair of equations. Time to call in the mathematician detectives to solve this mystery!
Remember, though, these answers may vary, depending on the equations you're working with. So, always double-check your equations before you jump to conclusions!
Stay curious, my friend!
To find the equilibrium quantity and price, we need to solve the system of equations:
2x + 9p - 93 = 0 (Equation 1)
3x - 14p + 108 = 0 (Equation 2)
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution:
From Equation 1, we can solve for x:
2x = 93 - 9p
x = (93 - 9p) / 2
Substitute this expression for x in Equation 2:
3((93 - 9p) / 2) - 14p + 108 = 0
Simplify and solve for p:
(279 - 27p) / 2 - 14p + 108 = 0
279 - 27p - 28p + 216 = 0
495 - 55p = 0
-55p = -495
p = (-495) / (-55)
p = 9
Now substitute the value of p back into Equation 1 to solve for x:
2x + 9(9) - 93 = 0
2x + 81 - 93 = 0
2x - 12 = 0
2x = 12
x = 6
Therefore, the equilibrium quantity is 6,000 units (6 * 1,000) and the equilibrium price is $9.
So the correct answer is d. equilibrium quantity 6,000 units; equilibrium price $9.
To find the equilibrium quantity and equilibrium price, we need to solve the system of equations:
2x + 9p - 93 = 0 ...(1)
3x - 14p + 108 = 0 ...(2)
There are a few different methods to solve this system of equations, but we'll use the method of substitution.
Let's solve equation (1) for x:
2x + 9p - 93 = 0
2x = 93 - 9p
x = (93 - 9p)/2 ...(3)
Now we substitute the value of x from equation (3) into equation (2):
3((93 - 9p)/2) - 14p + 108 = 0
Multiply through by 2 to eliminate the fraction:
3(93 - 9p) - 28p + 216 = 0
279 - 27p - 28p + 216 = 0
495 - 55p = 0
-55p = -495
p = 9
Now substitute the value of p = 9 back into equation (3) to find x:
x = (93 - 9(9))/2
x = 30
So, the equilibrium quantity is 30 units (30,000 in terms of 1,000 units), and the equilibrium price is $9.
Therefore, the correct answer is (d) equilibrium quantity 6,000 units; equilibrium price $9.