given the demand equation 8x + p - 46 = 0 and the supply equation 7x - p + 18 = 0, where p is the unit price in dollars and x represents the quantity demanded in units of a thousand, determine the equilibrium quantity and equilibrium price.
To find the equilibrium quantity and equilibrium price, we need to solve the given demand and supply equations simultaneously.
The demand equation is 8x + p - 46 = 0.
The supply equation is 7x - p + 18 = 0.
Step 1: Solve for p in terms of x by rearranging the supply equation.
7x - p + 18 = 0
p = 7x + 18
Step 2: Substitute the value of p from the supply equation into the demand equation.
8x + (7x + 18) - 46 = 0
8x + 7x + 18 - 46 = 0
15x - 28 = 0
Step 3: Solve for x by rearranging the equation.
15x - 28 = 0
15x = 28
x = 28/15
Step 4: Substitute the value of x back into the supply equation to find p.
p = 7x + 18
p = 7(28/15) + 18
p = 196/15 + 18
p = (196 + 270)/15
p = 466/15
Therefore, the equilibrium quantity is x = 28/15 (approximately 1.87 thousand units), and the equilibrium price is p = 466/15 (approximately $31.07).