Question 1202814
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.