A shoe store owner will buy 10 pairs of a certain shoe if the price is $79 per pair and 30 pairs if the price is $29. The supplier of the shoes is willing to provide 35 pairs if the price is $84 per pair but only 5 pairs if the price is $24. Assuming the supply and demand functions for the shoes are linear, find the market equilibrium point.

(54,20)

Step-by-step explanation:

From the data given, we need to calculate the slope of the line representing the demand and then determine the equation of the line.

we can split the demand points into

(quantity, price) hence we have

(10,30)(79,29)

Using the gradient equation

Since the equation of a line is define as

y=mx+c, we need to define the value of c

we substitute when y=30 and x=29

30=-(2/5)29+c

c=30+11.6

c=41.6

Hence the required demand line equation is

y=-0.4x+41.6

We also need to find the supply line equation,

we form the pairs from the given data

at $84 the supply is 34 pairs, at $24 the supply is 5 pairs

hence (34,5) and (84,24).

Next we determine the gradient

Next we determine the slope

we substitute when y=5 and x=24

5=(1/2)24+c

c=5-12

c=-7

Hence the required supply line equation is

y=0.5x-7

Next by equating both equation, we can determine the equilibrium price and quantity

for

-0.4x+41.6=0.5x-7

-0.4x-0.5x=-7-41.6

-0.9x=-48.6

x=54

substitute x=54 into both equation to get the same value for y

y=0.5(54)-7

y=20

also

y=-0.4x+41.6

y=-0.4(54)+41.6

y=-21.6+41.6

y=20

Hence the equilibrium point is (54,20)

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