 Alright, so now we're going to take a look at linear applications. When we're talking about linear applications, we have independent variable and dependent variable. Honestly, I don't use those terms very often, but I do talk about the input and the output. The independent variable usually represents the input. It's not dependent on anything. That's what you need to know to be able to find something else. The dependent variable is the output, and very often it is, if you read the problem, what they're asking you to find will be the output, not always, but very often. So if I had a problem that asked me the height h of a child that is two years old, if I want to think about the dependent variable, it's asking me to find the height of a child that's two years old. So the dependent variable would be the height. So let's look at an example. A doctor has $5,000 in fixed costs, so that's the piece of information we know, and that will cover her rent, salaries, equipment, and utilities. For each patient she sees, so it's for each patient, she has an additional cost of $15. And then the doctor charges $80 for an office visit. So that's the information that will be important to us today. So let's look at this problem. It asks us to write an equation for the total monthly cost when V patients visit are done in a month. And we want to know the dependent variable, or what are we trying to solve for? Okay, they're asking us to find costs, so that would be the dependent variable, and what's cost equal to? We'll go back up and look at it. So costs are $5,000 in fixed costs, and they're fixed, so there would be no variable to it. Every month they're going to pay $5,000. Plus, they're going to have to do this additional cost of $15 per patient. So we're additional $15 per visit, so that would be times V. So that would be our cost equation. Total monthly costs, now that we know what that was. And let's remind ourselves, it was C was equal to $5,000 fixed plus $15 per visit. And we want to know what the cost will be if we have 100 patient visits in a month. So that means that V is going to be 100. Patient visits is 100, so we say cost. We plug and chug $5,000 plus our 15 times our V, which is 100. And then we have $5,000 plus $1,500. So cost is going to equal $6,500 if they have 100 patients in a month. Consider this, if the doctor pays $6,800 in monthly costs, how many visits occurred? So now we know $6,800 is a cost. And we want to find the visits. This is what we're looking for, is the visits. Same equation, C is equal to $5,000 because it's still a cost equation, plus 15 V, but this time we know C. So $6,800 goes where the cost is, and then we finish out the rest of the equation. So with linear equations, remember we want to take the constants and put them on one side. So we're going to subtract the $5,000 from both sides. And that'll give us $1,800 is equal to 15 V. And if we divide everything by 15, then we're going to find out that V is equal to 120, or 120 patient visits per month. All right, now let's talk about some terms here. Revenue is considered the amount that you earn, the money you bring in. So they're asking us what monthly revenue for V patients in a month. And that's where this last sentence, we finally get to this last sentence, says Dr. charges $80 per office visit. So revenue is going to be equal to $80 per visit. That's how they bring in money is by charging the patients. Using that equation then, we should be able to answer some questions. Okay, they're going to ask us now. So we want to know profit now. Profit is equal to the revenue minus the cost. Remember, our revenue that we just found was 80 V. And the cost equation that we found earlier was cost was equal to 5,000 plus 15 V. So if I want a profit equation, it's going to be equal to the revenue, which is 80 V. And then we're going to subtract from that this whole cost. We're not just subtracting 5,000, we're subtracting 5,000 plus 15 V. So you got to distribute it in here so that we end up with P is equal to 80 V minus 5,000 minus 15 V. Or P is going to be equal to 65 V, 80 minus 15 V minus 5,000. This is our profit equation. So now let's answer the question. How much profit is made in a month for 150 visits? So V is equal to 150. So P is equal to 65 times 150 minus 5,000. And 65 times 150 happens to be 9,750, 750 minus 5,000. So profit is equal to, subtracting that we would have $4,750 for 150 visits. So how many patient visits must the doctor have in order to make a profit of 1,500? Okay so profit we said was equal to 65 V minus 5,000. And now we know that P is equal to 1,500. So we're going to have 1,500 is equal to 65 V minus 5,000. And if we add 5,000 to both sides, this will be 6,500 is equal to 65 V. And if we divide by 65, that will give us 100. So that would be 100 visits for a profit of $1,500. Alright final question. How many patient visits must the doctor have in order to break even? The break even point is when the revenue is equal to the cost. We're not making any money but we're not losing any money either. Well that's revenue. And remember revenue is equal to 80 V. And we have cost is equal to, and if we go back and remember earlier, the cost for the $5,000 fixed plus $15 per patient visit. So break even is going to be the revenue equal to the cost. So 80 V equal to 5,000 plus 15 V. Again getting our constants on one side variables on the other. The 15 V needs to go to the other side. And we're going to have 65 V is equal to 5,000. Dividing by 65 then is going to give us an amount of 76.9. So we'll say it's approximately 77 visits to break even.