Question 1194481: At a price of $10.96 per pound, the supply of beef ribeye is 354 thousand pounds and the demand is 405 thousand pounds. At a price of $12.52, the supply of beef ribeye is 415 thousand pounds and the demand is 344 thousand pounds.
(a) Find a price-supply equation of the form p=mx+b, where x is the quantity in thousands of pounds. Since one side of the equation has already been provided, you may provide your answer below is a constant times x plus a constant, or any algebraically equivalent expression.
p=
(b) Find a price-demand equation of the form p=mx+b, where x is the quantity in thousands of pounds.
p=
(c) What is the equilibrium quantity?
thousand pounds
(d) What is the equilibrium price?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you let x = the number of pounds of beef and y = the price per pound of beef, then you get the following points in (x,y) format.
supply points are (354000,10.96) and (415000,12.52)
demand points are (405000,10.96) and (344,000,12.52)
the linear equation is in the form of y = mx + b
m is the slope
b is the y-intercept.
the slope in each equation is (y2-y1) / (x2-x1)
for the demand equation, the slope becomes (12.52 - 10.96) / (344000 - 405000) which is equal to 1.56 / -61000 which can be shown as -1.56/61000.
for the supply equation, the slope becomes (12.52 - 10.96) / (415000 - 354000) which is equal to 1.56 / 61000 which can be shown as 1.56/61000.
the demand equation becomes y = -1.56/61000 + b
the supply equation becomes y = 1.56/61000 + b
to solve for b in each equation, take one of the points in each equation and use that point to find b.
in the demand equation, y = -1.56/61000 * x + b becomes 10.96 = -1.56/61000 * 405000 + b.
solve for b to get:
b = 10.96 + 1.56/61000*405000 = 21.31737705.
in the supply equation, y = 1.56/61000 * x + b becomes 10.96 = 1.56/61000 * 354000 + b.
solve for b to get:
b = 10.96 - 1.56/61000 * 354000 = 1.906885246.
your demand equation becomes y = -1.56/61000 * x + 21.31737705.
your supply equation becomes y = 1.56/61000 * x + 1.906885246.
in these equations, x represents the number of pounds of beef and y represents the price per pound.
y and p are interchangeable, since they both represent the same thing.
using p, you get:
your demand equation becomes p = -1.56/61000 * x + 21.31737705.
your supply equation becomes p = 1.56/61000 * x + 1.906885246.
y replaces p for graphing purposes only.
the equilibrium point is when the demand is equal to the supply.
this occurs when the pounds of beef are the same for the supply and for the demand.
to find that point, set the equations equal to each other and solve for x.
you get:
-1.56/61000 * x + 21.31737705 = 1.56/61000 * x + 1.906885246.
add 1.56/61000 to both sides of the equation and subtract 1.906885246 from both sides of the equation to get:
3.12/61000 * x = 19.4104918
solve for x to get:
x = 19.4104918 * 61000 / 3.12 = 379500.
the supply and demand will be in equilibrium when 379,500 pounds of beef are supplied and demanded.
when that happens, the price per pound of beef will be equal to 11.61213115.
i graphed both equations.
red is the demand equation.
blue is the supply equation.
the graph looks like this:
answers to your questions are shown below.
(a) Find a price-supply equation of the form p=mx+b, where x is the quantity in thousands of pounds. Since one side of the equation has already been provided, you may provide your answer below is a constant times x plus a constant, or any algebraically equivalent expression.
p = 1.56/61000 * x + 1.906885246
(b) Find a price-demand equation of the form p=mx+b, where x is the quantity in thousands of pounds.
p = -1.56/61000 * x + 21.31737705.
(c) What is the equilibrium quantity?
379.5 thousand pounds (379,500).
(d) What is the equilibrium price?
11.61213115 dollars per pound.
let me know is you have any questions.
theo
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