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Economists work with supply and demand curves that show the price \(\mathrm{P}\) of goods as a function of the quantity \(\mathrm{Q}\) of those goods supplied or demanded. For a certain product, the supply curve is \(\mathrm{P = aQ + b}\) and the demand curve is \(\mathrm{P = \frac{k}{Q}}\), where \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{k}\) are nonzero constants. The point at which these curves intersect in the \(\mathrm{(Q, P)}\) coordinate plane is referred to as the equilibrium point, and for this product the equilibrium point is \(\mathrm{(10,5)}\). For this product, what are the values of \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{k}\)?
We need to find the specific values of three constants: a, b, and k.
For sufficiency, we must be able to calculate the exact numerical values of a, b, and k.
Since both curves pass through the equilibrium point \(\mathrm{(10, 5)}\):
Notice: We already know k! But we have one equation with two unknowns (a and b). We need one more independent equation to solve for both values.
Statement 1: The point \(\mathrm{(8, 1)}\) is on the supply curve.
If \(\mathrm{(8, 1)}\) is on the supply curve \(\mathrm{P = aQ + b}\), then:
\(\mathrm{1 = 8a + b}\)
Now we have two equations:
Subtracting the second equation from the first:
Substituting \(\mathrm{a = 2}\) back into either equation:
We can determine all three values: \(\mathrm{a = 2, b = -15, k = 50}\).
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The point \(\mathrm{(25, 2)}\) is on the demand curve.
If \(\mathrm{(25, 2)}\) is on the demand curve \(\mathrm{P = k/Q}\), then:
\(\mathrm{2 = k/25}\)
This gives us \(\mathrm{k = 50}\).
But wait—we already knew \(\mathrm{k = 50}\) from the equilibrium point! Statement 2 merely confirms what we already had. It provides no new information.
We still have only one equation for a and b:
With one equation and two unknowns, we cannot determine unique values for a and b. For example:
Statement 2 is NOT sufficient.
This eliminates choices B and D.
Since Statement 1 alone is sufficient to determine all three values, but Statement 2 alone is not sufficient, the answer is A.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."