SOLUTION: The Unique Gifts catalog lists a "super loud and vibrating alarm clock." Their records indicate the following information on the relation of monthly supply and demand quantities to

Question 1202814: The Unique Gifts catalog lists a "super loud and vibrating alarm clock." Their records indicate the following information on the relation of monthly supply and demand quantities to the price of the clock.
Demand Supply Price
166 131 $34
146 181 $42
Use this information to find the following.
(a) points on the demand linear equation
(x, p) =( ) (smaller x-value)
(x, p) =( ) (larger x-value)

points on the supply linear equation
(x, p) =( )(smaller x-value)
(x, p) =( )(larger x-value)

(b) the demand equation p
p =
(c) the supply equation p
p =

(d) the equilibrium quantity and price
Equilibrium occurs when the price of the clock is $________ and the quantity is _________.
Found 2 solutions by jekoishun, greenestamps:
Answer by jekoishun(6) (Show Source): You can put this solution on YOUR website!
You can find the equations for both demand and supply using the information provided. First, we need to find the slope and the y-intercept for both the demand and the supply lines.
(a) The points on the demand linear equation are given by the pairs of quantity demanded and price:
$ (x, p) = (166, 34) $ (smaller x-value)
$ (x, p) = (146, 42) $ (larger x-value)
The points on the supply linear equation are given by the pairs of quantity supplied and price:
$ (x, p) = (131, 34) $ (smaller x-value)
$ (x, p) = (181, 42) $ (larger x-value)
(b) To find the demand equation, we will use the two points on the demand line:
$ (166, 34) $ and $ (146, 42) $
The slope of the demand line can be found using:
\[ m = \frac{{42 - 34}}{{146 - 166}} = \frac{{8}}{{-20}} = -\frac{2}{5} \]
We can use one of the points to find the y-intercept b:
\[ 34 = -\frac{2}{5} \cdot 166 + b \]
\[ b = \frac{2 \cdot 166}{5} + 34 = \frac{332}{5} + 34 = 66.4 + 34 = 100.4 \]
So the demand equation is:
\[ p = -\frac{2}{5}x + 100.4 \]
(c) Similarly, for the supply equation using the points $ (131, 34) $ and $ (181, 42) $, the slope is:
\[ m = \frac{{42 - 34}}{{181 - 131}} = \frac{{8}}{{50}} = \frac{2}{5} \]
And the y-intercept can be found as:
\[ 34 = \frac{2}{5} \cdot 131 + b \]
\[ b = \frac{2 \cdot 131}{5} + 34 = \frac{262}{5} + 34 = 52.4 + 34 = 86.4 \]
So the supply equation is:
\[ p = \frac{2}{5}x + 86.4 \]
(d) The equilibrium quantity and price are found where the demand and supply equations intersect:
\[ -\frac{2}{5}x + 100.4 = \frac{2}{5}x + 86.4 \]
Combining like terms:
\[ -\frac{4}{5}x = -14 \]
\[ x = \frac{14 \cdot 5}{4} = 17.5 \]
Substituting back to find the price:
\[ p = -\frac{2}{5} \cdot 17.5 + 100.4 = -7 + 100.4 = 93.4 \]
So the equilibrium occurs when the price of the clock is $93.4, and the quantity is 17.5.
Note: It seems like there may be an inconsistency in the given data, as the demand equation suggests a positive price for negative quantities and the supply equation suggests a positive price for quantities greater than 181. Make sure to double-check the provided data and the context in which these equations are being used.
Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

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Suggestions for new tutor @jekoishun...

(1) Check your work and your answer before submitting your response. Your response to this question shows that you know how to solve the problem... but somewhere along the way you used some wrong numbers and ended up with a wrong answer.

(2) Look at "recently solved" and click on "show source" to learn how to make your responses much more readable using html code. Or you can find some good tutorials on html on the internet.

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I will provide an outline of the process and let the reader fill in the details; showing the complete solution does not benefit the student....

Parts a, b, and c...

For the demand equation the two points are (x,p) = (166,34) and (146,42). Use basic algebra to find the demand equation is

For the supply equation the two points are (x,p) = (131,34) and (181,42). Use basic algebra to find the supply equation is

Part d...

Set the expressions for p in the demand and supply equations equal and solve to find the equilibrium point.

...

You should find the equilibrium point is (x,p) = (156,38)

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