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 demand.

Accepted Answer

Hello everybody. I have everything right today. Today, we're gonna be looking at this math question that states find the equilibrium demand using the following supply and demand equations where the supply equation is 1000 divided by 4000 minus Q and the demand equation is 14, minus five Q and all that divided by four Q. Now the answers provided are a 1000 B, 5600 C 2000 and D 4000. Now we see that they're asking for the equilibrium demand. So when do we achieve equilibrium? When we have, when we're talking about supply and demand, we can recall that equilibrium is achieved when supply is equal to demand. So when you have the same supply and demand, you will reach a level of equilibrium. So to get our value, we can set our supply equation equal to our demand equation. So we'll have 1000 divided by 4000 minus Q and that's equal to 14,000 minus five Q and all that divided by four Q. Now it's just a matter of stopping for queue. The queue here will represent the number of items for the demand. So what we can do now is start simplifying. So the denominator on the left hand side, the 4000 minus Q can be brought over to the right hand side by multiplying it. And the denominator on the right hand, right hand side, the four Q can be brought over to the left hand side by multiplying it as well. So on the left hand side, we'll have 1000 multiplied by four Q and that'll be equal to 14,000 minus five Q and all that multiplied by minus Q. Now we have two multiplications on each side which we can distribute the values and simplify a little further. So on the left hand side, we have 1000 multiplied by four Q which will be 4000 Q and that'll be equal to. And on the right hand side, it's a little bit more complicated distribution. So let's just go little by little. First, we have 14,000 multiplied by 4000 which is 56 million. And then we would have 14,000 multiplied by the negative Q which is negative 14, Q. Then we have our negative five Q multiplied by the 4000 which will be negative 20,000 Q. And finally, we have our negative five Q multiplied by the negative Q which will be positive five Q squared. So in total, we have 4000 Q equals 56 million minus 14,000 Q minus 20,000 Q plus five Q squared. So let's do a little bit more simplification. We see that we have two terms with Q subtracting each other. So let's simplify that. So we'll have 4000 Q is equal to 56 million. And now we had negative 14,000 Q minus 20,000 Q which will be negative 34 Q and that'll still be plus five Q squared. So now we can do a little bit of rearranging as well as bringing over the 4000 cue from the left hand side to the right hand side to have this equation equal to zero. So we will have zero on the left hand side and that's equal to and I'll do some rearranging. So I'll put the Q squared term first. So that, and then, well, I'll put the Q squared term first, then I'll put the Q term second and the number last. So it's in a more of a way that is clear where we're going afterwards, which will be factoring. So that's so this is a, a way that we can see it more clearly. So we have zero is equal to five Q squared and then we had minus 34,000 Q, but now we also subtracted the 4000 Q from the left hand side. So it'll be minus 38,000 Q and we still have plus 56 million. So now we have this in a more clear factoring terminology and we can do exactly that factor. So the first thing I want to factor out is a five. I see that every term here, the five Q squared, the 38,000 and the 56 million. I see that I can factor out five out of all of them. So that's exactly what I'm gonna do. So I'll have, I'll factor out of five, so it'll be five and then inside of the parentheses, so it will be five multiplied by. And then I'll have Q squared minus Q plus 11,200,000. And now I can factor this further. So to factor this, I gotta come up with uh two parentheses that will multiply each other where the, I'll have a queue as the first term in both parentheses. And then I need to find two numbers that when multiplied by each other will give me 11,200,000. But when added will give me negative 7600. So those two numbers are what we had initially. So the 7600 Q is what we had as the middle term and the 11,200,000 was the last term in our equation. So those two numbers would be negative 2000. So we'll have Q minus 2001 parentheses. And then the other parentheses we'll have Q minus and we'll have 5600. So you notice that both of those numbers are negative because when multiplied by each other, they need to give us a positive 11 minutes on 1000. And then when we added, when added to each other, they will be the negative 7600 that we needed and that will still be equal to zero. So we have five and then multiplying one parentheses, which is Q minus 202 minus 2000 and multiplying another parentheses which is Q minus 5600. So now we can set the queue or each parenthesis equal to zero. So Q minus 2000 equals zero, N Q minus 5600 equals zero. And this is because to obtain the answer of zero, one of the parentheses can be zero. And since everything is multiplying as long as one parenthesis is zero, then we will equal zero at the end. So that's all we need. Now. So now we solve for Q so we had Q minus 2000 0. So have Q is equal to positive 2000. And then we also had Q minus 5006, 600 equals zero, which is Q equals five positive 5600. Now we have two answers for A Q but we only need one and we can only use one. So we look at both answers the 5600. When we plug it back into our supply and demand equations, we see that it'll give us a negative answer and that's just not possible. You cannot have a po a negative supply or a negative demand. So we have to eliminate the 5600 which means we are left with Q equals 2000 in that queue. Like I mentioned earlier will represent the amount of items found in the equilibrium demand. So that'll be answer choice C 2000. 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 demand.

Key Concepts

Equilibrium in Supply and Demand

Solving Rational Equations

Interpreting Variables in Context

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