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In the last video we began to visually explore the relationship between the number of sessions I attend and my monthly cost given the different plans.

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So right here, in this blue line we visualized it for the trial plan, and then this orange dotted line we visualized it for the basic plan.

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And it meets where it intuitively, hopefully represents what we already understood about the different plans.

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The trial plan, we don't pay anything upfront and every time we add a session, so every time we move to the right here we move to the right one session.

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We add twelve dollars to out cost, we move up by twelve.

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Add another session, move up by 12, add another session move up by 12.

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On the basic plan we did have an upfront cost on 20 dollars

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but the line was less steep, we had to pay less for each extra session.

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Here when we added an extra session we only had to add 8 dollars to our total fees.

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Add another session, you only have to add 8 dollars.

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So the orange line starts at a higher point but it is less steep, and we see because of that they clearly intersect.

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And they intersect right over here.

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So my question to you is, "What is the significance of that point of intersection?"

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Well lets think about it a little bit.

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How much do each of the plans cost if I only attend one session?

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So we see here from the trial plan it will cost us 12 dollars while on the basic plan one session will cost us 28 dollars.

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And you don't even have to look at these tables we set up. You can see this visually.

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So if we go up one session the trial plan is below the basic plan.

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If you go up 2 sessions the trial plan is still below the basic plan.

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Although they are not as far apart.

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Three sessions, still the basic plan is above the trial plan but they are getting closer and closer together.

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All the way until you get to this point over here.

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This point is essentially the number of sessions, it looks like five sessions,

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it looks like the number of sessions regardless of the plan we choose the cost is the same.

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It looks like that cost is right around 60 dollars.

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Once we go beyond that point, all of a sudden the basic plan becomes cheaper than the trial plan.

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It looks like if we were to take 6 sessions the basic plan would give a lower price than the trial plan.

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But how do we actually figure out what this point is, and what number of sessions, and what dollar value?

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I just eyeballed it right now which is useful with these visual graphs.

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But what if we wanted to get the exact value and the value is 5.1 or 60.25

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How do we get the exact same value?

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Well, one way to think about it is, were trying to find out s, or the number of sessions, so that regardless of which plan we choose we have the exact same cost.

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So if we pick the basic plan our cost is going to be 20 dollars plus 8 times s.

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And if we pick the trial plan the cost is going to be 12 times s

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And were trying to find that unknown s where this value is going to be the same as this value.

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where that unknown value of sessions, where regardless of which plan i choose i will pay the same price

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so we're curious about the s where 12s is equal to 20 plus 8s.

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So we set up this equation.

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the equation has one unknown and we should be able to solve this.

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So lets think about how we can do it.

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What is the first general idea if we want to figure out the unknown s.

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There are many ways to approach theses Algebra problems which is what makes them fun

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I try to isolate the s on one side of the equation.

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For me, i want to isolate it on the right side of the equation.

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Since i already has this 12s there and this 20 on the left side.

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I could have done other things.

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So what is the best way to get rid of this 8s from the left side.

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Well the easiest thing i can think of is subtract 8s from lefthand side

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But i can't just do that, if I do then this equality won't hold.

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These things were equal to each other and if i just subtract from one side then the left side wont be equal to the right side

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In order for them to be equal i have to do the same thing on the right side

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Once I do that on the left side these two characters negate each other and i am left with 20 on the left side.

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And on the right i am left with 12s and take away 8s and i am left with 4s.

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So it is equal to 4 times s.

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And so now we are pretty close. I just want an s on the right side.

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What can I do to this equation, so i can have just an s on the righthand side.

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Well the easiest thing to do is divide both sides by 4

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I can't do it just to one side.

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When i divide both sides by 4, what do i get for s?

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Well on the right hand side 4s divided by 4 is just going to be s.

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And thats going to be equal to 20 divided by 4 which is 5

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So we eyeballed it and it looked like 5 sessions.

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Now we know for sure that at 5 sessions the cost of either plan is going to be the same.

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But what is the cost of either plan?

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Well we should be able to go to either plan because the cost will be the same.

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So is we look at out trial plan and we say 5 sessions, how much will that cost?

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well the cost is going to be 12 times 5

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so the cost is going to be equal to 12,12 times 5 which is going to be 60 dollars

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the cost is going to be 60 dollars.

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So my question to you is do we even have to try out the 5 in this equation right over here?

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See what it would cost under the basic plan.

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Well we wouldn't because the whole reason we got 5 sessions is because we said this is the number f sessions where we get the same cost

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For the basic plan or the trial plan

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but if you're curious i do encourage you to find out.

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Substitute a 5 in for s and see that you get

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you'll get 20 plus 8 time 5 which is 20 plus 40 which is 60 dollars

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So with either plan at 5 sessions the total cost is going to be 60 dollars.

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Then is you add sessions after that,

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Because the trial plan is more steep each incremental session after that is going to cost you more.

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You start to see that the trial plan begins to get more and more expensive.

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So going back to my question of which gym membership i'm going to use.

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All of a sudden i have an interesting answer

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If i plan on attending on average less that 5 sessions

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it probably make sense, less that 5 sessions a month, it makes sense for me to use the trial plan.

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If i plan to attend more that 5 sessions a month the basic plan is going to be cheaper.

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If i plan to attend exactly 5 sessions a month it doesn't matter which plan i acutally use