Q as a function of P P as a function of Q Q = 30 – P Q = 12 – 3P 6Q = 14 – 2P
The Correct Answer and Explanation is:
Let’s find:
- Q as a function of P
- P as a function of Q
We’ll do this for each equation:
Equation 1: Q = 30 – P
Q as a function of P: Already given as
Q(P) = 30 – P
P as a function of Q:
Solve for P:
Q = 30 – P
=> P = 30 – Q
P(Q) = 30 – Q
Equation 2: Q = 12 – 3P
Q as a function of P: Already given as
Q(P) = 12 – 3P
P as a function of Q:
Q = 12 – 3P
=> 3P = 12 – Q
=> P = (12 – Q) / 3
P(Q) = (12 – Q) / 3
Equation 3: 6Q = 14 – 2P
Solve for Q:
6Q = 14 – 2P
=> Q = (14 – 2P) / 6
=> Q = (7 – P) / 3
Q(P) = (7 – P) / 3
Now solve for P:
6Q = 14 – 2P
=> 2P = 14 – 6Q
=> P = (14 – 6Q) / 2
=> P = 7 – 3Q
P(Q) = 7 – 3Q
Explanation
In algebra, expressing one variable in terms of another helps us understand the relationship between variables. This is commonly done in economics and mathematics when working with demand and supply functions, where quantity Q depends on price P or vice versa.
For the first equation Q = 30 – P, Q is already expressed as a function of P. Rearranging it to express P as a function of Q gives P = 30 – Q. This simply reverses the dependency, showing how price changes when quantity changes.
In the second equation Q = 12 – 3P, again Q is given as a function of P. Solving for P means isolating P on one side. We do this by subtracting Q from 12 and then dividing by 3, resulting in P = (12 – Q) / 3. This is essential when we want to determine the price needed to achieve a certain quantity.
The third equation, 6Q = 14 – 2P, initially shows a relationship between Q and P that is not directly solved for either variable. We must manipulate the equation by isolating Q or P to convert it into a function form. Solving for Q gives Q = (7 – P) / 3, and solving for P gives P = 7 – 3Q. These forms are useful in applications such as analyzing how supply or demand shifts with changes in price or quantity.
Converting between Q(P) and P(Q) allows for better flexibility in mathematical modeling. It is especially important when plotting graphs or optimizing functions, as it allows a clearer view of how one variable responds to changes in the other.
