Question

A group of retailers will buy 104 televisions from a wholesaler if the price is $300 and 144 if the price is $250. The wholesaler is willing to supply 88 if the price is $225 and 168 if the price is $315. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market.

Answer 1

Answer:

(275,122)  

When the price will be $275, the quantity will be 122 televisions.

Step-by-step explanation:

We have been given that a group of retailers will buy 104 televisions from a wholesaler if the price is $300 and 144 if the price is $250.

As the quantity of televisions depends on price of televisions, so our demand curve will pass through points (300,104) and (250,144).

Let us find slope of demand line using slope formula.

$m=\frac{y_2-y_1}{x_2-x_1}$, where,

$m =\text{Slope of line}$,

$y_2-y_1=\text{Difference between two y-coordinates}$,

$x_2-x_1=\text{Difference between same x-coordinates of two y-coordinates}$

Upon substituting coordinates of our given points in slope formula we will get,

$m=\frac{104-144}{300-250}$

$m=\frac{-40}{50}$

$m=-\frac{4}{5}$

Let us substitute $m=-\frac{4}{5}$ coordinates of point (250,144) in slope intercept form of equation ($y=mx+b$).

$144=-\frac{4}{5}*250+b$

$144=-4*50+b$

$144=-200+b$

$144+200=-200+200+b$

$344=b$

Upon substituting $b=344$ and $m=-\frac{4}{5}$ we will get equation of our demand line as:

$y=-\frac{4}{5}x+344$

Similarly we will find the equation of supply line using points (225,88) and (315,168).  

$m=\frac{168-88}{315-225}$

$m=\frac{80}{90}$

$m=\frac{8}{9}$

Let us substitute $m=\frac{8}{9}$ and coordinates of point (225,88) in slope intercept form of equation ($y=mx+b$).

$88=\frac{8}{9}*225+b$

$88=8*25+b$

$88=200+b$

$88-200=200-200+b$

$-112=b$

Upon substituting $b=-112$ and $m=\frac{8}{9}$ we will get equation of our supply line as:

$y=\frac{8}{9}x-122$

Let us equate both lines to find the point where both lines intersect.

$-\frac{4}{5}x+344=\frac{8}{9}x-122$

$-\frac{4}{5}x-\frac{8}{9}x+344=\frac{8}{9}x-\frac{8}{9}x-122$

$-\frac{4}{5}x-\frac{8}{9}x+344=-122$

$-\frac{4}{5}x-\frac{8}{9}x+344-344=-122-344$

$-\frac{4}{5}x-\frac{8}{9}x=-466$

Let us have a common denominator.

$-\frac{4*9}{5*9}x-\frac{8*5}{9*5}x=-466$

$-\frac{36}{45}x-\frac{40}{45}x=-466$

$\frac{-36-40}{45}x=-466$

$\frac{-76}{45}x=-466$

$\frac{-76}{45}*\frac{45}{-76}x=-466*\frac{45}{-76}$

$x=466*\frac{45}{76}$

$x=6.1315789473684211*45$

$x=275.9210526315789495\approx 275$

Let us substitute x=275 in any of our equation to solve for y.

$y=-\frac{4}{5}*275+344$

$y=-4*55+344$

$y=-220+344$

$y=122$

Therefore, the equilibrium point for the market will be (275,122).