Solve each problem. The supply and demand equations for a cert...

Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q.

Find the equilibrium price (in dollars).

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're gonna be looking at this math question that states find the equilibrium price in dollars using the following supply and demand equations where the supply equation is 1000 divided by 4000 minus Q. And the demand equation is 14,000 minus five Q and all of that divided by four Q. And the other choices that they give us are a $0.5 B $5 C $1 and D two daughters. Now, when looking at this equate this problem, I see that they're asking us specifically for the equilibrium price. So what does that mean? When do we achieve equilibrium? When we have supply and demand? Well, we know that equilibrium is achieved when the input is the same as the output or in this case, supply is equal to demand. So when the supply and the demand are equal to each other, you reach a sense of equilibrium. So we can set our supply equation equal to our demand equation. And that's exactly what we're gonna do. So we'll have 1000 divided by 4000 minus Q equal to 14,000 minus five Q and all of that divided by four Q. And now it's a matter of solving for Q. And to do that, we'll multiply the denominator on the left hand side. In this case, 4000 minus Q, we'll multiply that with the right hand side and the denominator of the right hand side. In this case, four Q will multiply that with the left hand side. So we'll have 1000 multiplied by four Q and that'll be equal to 14,000 minus five Q multiplied by 4000 minus Q. And now we can distribute. So on the left hand side, we will have 1000 multiplied by four Q which is Q. And on the right hand side, we have a little bit more complicated distribution. So let's go through it little by little. So first we'll have the 14,000 multiplied by the which that will be 56 million. And then we'll have the 14,000 multiplying the minus Q or a negative Q which is the same as saying negative 14,000 Q or minus 14 1000 Q. Then we'll have the minus five Q or negative five Q multiplying the 4000 which will be a negative 20, Q. So at that at minus 20,000 cube, and finally, we'll have the negative five Q multiplying the negative Q which is a positive or plus five Q squared, the variable will be squared. In this case, now, we can do a bit of simplification. So we had to see that we have a minus 14,000 Q and or a negative 14,000 minus 14,000 Q minus 20,000 Q. So we can subtract those and we have 14, 4000 Q equals million minus 34 Q plus five Q squared. Now, we can do some rearranging of our equation as well as bring over the 4000 Q from the left hand side to the right hand side to have a little bit more simplified equation as well. The rearranging will be rearranging it to have the Q squared first, the Q second and the number last. So if we bring over the 4000 cube to the right hand side, that will leave us with zero on the left hand side and we'll have five Q squared first is how we will rearrange it. Then we'll have the Q term, which was, in this case, it was negative, it was minus 34,000 Q, but we're subtracting another 4000 Q. So now we'll have minus 38,000 Q and finally, we'll have plus million. And now we can see that this is just a simple factoring problem and that was why rearranged it this way. What we have the Q squared first, the Q second and the number last. It's a better way of seeing how we can factor. It's what we're accustomed to So the first thing I'm gonna factor out here is a five and you see that every single term has a five in it or I can factor out a five. So we have five factored out and that will leave us with a five multiplied by Q squared minus 7600 Q plus 11,200,000. And that is still equal to zero. So now I can factor out, continue to factor the, the, what's inside the parentheses, which in, in this case is a Q squared minus 7, 7600 Q plus 11,200,000. And to factor that I have to see what two numbers multiplied by each other will give me the number term, the 11,200,000. And when you add them up, it gives you the middle term. In this case, a negative 7600 Q. So we'll have a five multiplying. And the first thing I'll factor out is the QS because I know we have Q squared, I know both parentheses. In this case, we'll have a Q to get the Q squared. And now the first term will be Q minus 2000 and the second term will be Q minus 5600. So we have, when you have Q multiplied by Q, that's a Q squared. And then you have Q multiplied by negative 5600. And then we'll have that minus the 2000 multiplied by the Q. And when you add those up, you get the ne the minus 7600 Q in our original equation. And when you multiply the negative 2000 multiplied by the negative 5600 you get a positive 11,200,000, which is the final number in our original equation. So this factor out perfectly. No, to figure out the values of Q, we're gonna have two separate values. We already factor out the five. So we don't have to worry about that. And now we'll have Q minus 2000 equals zero or Q minus 5600 equal to zero. You set both equal to zero because as long as any of the parentheses are zero, we will get uh an answer of zero since they're multiplying each other. So we'll get one answer of Q as positive 2000 and another as Q equals positive 5600. Now, when we look back at, at our original equation, if we plug in the 5600 we will get a negative answer, which is just not possible. In this case, you can't have a negative supply or a negative demand. So we can eliminate the 5600 and stick with Q equals 2000. Now, we can plug that back into our supply equation to figure out the price. So we'll have price or P equals 1000 divided by 4000. Minus Q which is equal to divided by 4000 minus 2000 and that is equal to 1000 divided by 2000, which is equal to one half. So ping or price is equal to one half or 0.5 and that is in dollars. And that will be your answer for the price at equilibrium. So half a dollar or answer choice A in this case. So I hope that was helpful. And until next time.

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q.
Find the equilibrium price (in dollars).

Key Concepts

Equilibrium Price and Quantity

Solving Rational Equations

Interpreting Supply and Demand Functions

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