Question

In the pair of supply and demand equations below, where x represents the quantity demanded in units of a thousand and p the unit price in dollars, find the equilibrium quantity and the equilibrium price. p = −x2 − 3x + 80 and p = 7x + 5

Answer 1

Answer:

Equilibrium quantity = 5

Equilibrium price = 40

Explanation:

given:

p = -x²-3x+80

p = 7x+5

For the equilibrium quantity the price from both the functions will be equal

thus, we have

-x² - 3x + 80 = 7x+5

⇒ x² +3x + 7x + 5 - 80 = 0

⇒x² + 10x - 75 = 0

now solving for x

x²- 5x + 15x -75 = 0

x(x-5) + 15(x-5) = 0

therefore, the two roots of the equation are

x = 5 and x = -15

since the quantity cannot be in negative

therefore, the equilibrium quantity will be = 5

now the equilibrium price can be found out by substituting the equilibrium quantity in any of the equation

thus,

p = -(5)² -3(5) + 80 = 40

or

p = 7(5) + 5 = 40

Answer 2

In this exercise we have to use the knowledge of finance to calculate the equilibrium value and the quantity, so we have:

• Equilibrium quantity = 5
• Equilibrium price = 40

Organizing the information given in the statement we have:

• p = -x²-3x+80
• p = 7x+5

So equating the two given equations we have:

$-x^2 - 3x + 80 = 7x+5\\x^2 +3x + 7x + 5 - 80 = 0\\x^2 + 10x - 75 = 0\\x^2- 5x + 15x -75 = 0\\x(x-5) + 15(x-5) = 0$

So we can see that the roots will be x = 5 and x = -15 since the quantity cannot be in negative therefore, the equilibrium quantity will be = 5 So replace that value at:

$p = -(5)^2-3(5) + 80 = 40\\p = 7(5) + 5 = 40$

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