hello college algebra students this video covers the second half of section 1.2 applications and modeling with linear equations when we first began this lesson we went over six steps for solving all applications and I want to review those very quickly now for step one I've got to read the problem carefully I've got to know what I've been given and I have to know what it is I have to find step two assign a variable there's always going to be an unknown value that's what I'm solving for and I need to assign it a variable what is a variable a letter so you can assign it as X you know that's the universal variable or t if you're solving for time or n whatever you choose but you need to assign a variable to your unknown and then we write an equation an equation that ties everything together we solve that equation using the steps that we learned in section one 1.1 and then we State our answer to the problem ask ourselves does it seem reasonable and then we will check to make sure the solution is accurate those are the steps let's look at example four how many gallons of a 25% anti-free solution should be added to five gallons of a 10% solution in order to obtain a 15% solution well when we first started this lesson I told you there are three types of problems that you're going to see geometry problems with which deal with length and width and rectangular areas and all of that perimeter all of that and then mixture problems you're going to see and also motion this is an example of a mixture problem so think about what's happening here I have a 10% solution and I want to bump it up to a 15% solution of antifreeze and I have a 25% so I'm going to mix the two the 25% and the 10% together in order for the 10% to be increased to 15% that's the idea of what's going on here let's look at our chart well similarly to what we had before I'm going I have my strength I have the gallons and then I have the gallons of pure antifreeze let's fill in what we have first let's take it line by line or even if necessary Clause by Clause how many gallons hopefully you should know just from those first three words that we don't know how many gallons that is our unknown so under gallons I'm going to put X how many gallons of a 25% anti-freeze okay how many gallons at strength 25 must be added to pay attention now five gallons well I put five in my gallon spot five gallons of a 10 % solution well let's put the 10% where it belongs in order 10% sorry in order to obtain a 15% Solution that's what we've been given that is the only thing that we've been given from the problem listen listen carefully how many of a 25% strength of antifreeze should be added to 5% of a 10% antifreeze in order to get 15% that's what we've been given from the problem itself now I need your full attention on this part here okay all of my toll totals are flush right and at the bottom whenever you have a mixture problem your totals are in the area that I've just highlighted okay when you are trying to fill in these values you always multiply so you multiply going across when you're trying to fill in these values you add so now you can't see my hand so I'm going to try and do this in a way that hopefully you can understand okay I multiply let me use a different color I multiply going this way and I add going this way so I add in columns and I multiply in the rows now what does that look like that essentially means that I can figure out what belongs in each spot If I multiply going across then .25 * X would put a 0.25x here and .1 * 5 is 0.5 so notice those two things that I just wrote in I wrote based on the fact that in a mixture prop problem I multiply going across well I add coming down so remember I am mixing these two together in order to add to get 0.15 well what happens when I add these hopefully you see that's just x + 5 so I multiply going across and I add coming down well that means means that two things are going to end up in this one cell and those two things have to be equal what are they the sum of these two and the product of those two so if I added these two together what would I get 25x plus 0.5 because I add in my columns but if I were to put these two in the Box I know I would need to multiply .15 * x + 5 which means that both of these must be equal so 0.25x plus 0.5 belongs in that cell as well as .15 * x + 5 and so those two quantities must be equal all right have a lot of writing in here it looks kind of messy so now I'm going to take all those things off and let's solve our equation well I know when I've got a number on the outside of a bracket it must distribute in so this gives me 0.25x plus + 0.55 is equal to15 * x + I believe that's 75 let's double check yeah 75 and now I know I need to combine my like terms terms so here are the values that have x's in them so I'm going to get those together on the left side of the equation 25x and of course since this 0.15x was positive I'm going to subtract it from both sides and this value right here 0.5 I'm going to subtract it and put it over on this side all right well. 25x minus1 15x is just1 X and of course 75 -.5 is25 and now in order to get that X by itself I need to divide by 0.1 so X is going to be. 25 divided by that 0.1 and when we do that we get 2.5 gallons and we're done if you have a question please write it down because at the very beginning of the next lesson I will further explain whatever you need explained so how many gallons of a 25% strength should I add to 5 gallons of a 10% Strength 2 and2 gallons if I do that then I will increase my 10% strength to 15% and we're done let's look at the next one last year Ryan earned a total of 784 in interest from two Investments he invested a total of 28,000 part at 2.4% and the rest at 3.1% how much did he invest at each at each rate well I need your full attention for this one notice how if you look at it closely it appears as though we might have two unknowns Ryan had two Investments he invested part of it at 2.4 and the rest at 3 .1 we don't know how much was invested at each of those rates so I need you to listen if you ever have two unknowns and a total then you make your first unknown X and you make your second unknown the total minus X if you ever have a total and two unknown then one of the unknowns is X and the other unknown is the total minus X pay attention you and I together have $30 I don't know how much you have you don't know how much I have but together we have $30 does it make sense then that whatever I have you have got to have 30 minus that how does does that work well if I've got $2 you've got 30 minus 2 or 28 if I have $5 then you have 30 minus 5 or 25 so I don't assign two unknowns I assign one unknown and the other would be the total minus it hopefully that makes sense now let's go back and take it one at a time last year Ryan earned a total of 784 from two Investments well where does my totals go I already told you the totals are always to the right and at the bottom so my total for this $784 worth of interest interest total at the bottom all right a total of 28,000 well that is also a total 28,000 so it goes in my total area at the bottom or to the right part of it at 2.4% part we don't know X part of it at 2.4% okay that's 24% so let me make sure I have an extra Zero part of it at 2.4% listen and the rest the total minus that X at 3.1% this is what we have how much did he invest at each rate this is what we've been given by the problem two totals part the rest and my rates in the last problem I told you exactly what to do when you have a mixture problem and you have cells that you need to fill take a second to think about what I said hopefully you remember that I said you multiply going across I have a missing column here at the bottom I don't need that though if I have one whole column that I can fill out so why don't I use this final column this final interest column to figure out exactly what goes there and how do I do that by multiplying going across so I have 024 time the X and 0.31 time the 28,000 - x okay and what do I do coming down well I know that the sum of these has to be equal to 784 remember add going down multiply going across so I've got 024 * X Plus 031 * 28,000 - x and that is equal to 784 hopefully you recognize this is just a regular linear equation that can very easily be solved when I have a value on the outside it gets distributed on the inside so that gives me24 X Plus 031 * 28,000 031 1 time oh wait is it time yeah * 28,000 868 minus 0 31x and that is equal to 784 all right let's combine our like terms I see an X here and an X there so I know I can subtract those wouldn't that just give me um oh a negative 07 X I'll double check let me double check right now so that I don't waste time 024 minus 031 yes a negative 007 get it 007 never mind plus 868 that's equal to 784 and now I know with this number right here I've got to subtract it from both sides so I'll go ahead and do that so that I've got a 0.07 X is equal to 784 minus 868 so I've got a 0.7x is equal to 784 minus 868 a 84 and the only thing that remains is dividing both sides by7 Final Answer 12,000 for x now let's go back and look at the problem in its entirety the question asked us how much did he invest at each rate well we solved for x and found out that X is 12,000 so 12,000 was invested at 2.4% how do I get the rest 28,000 minus that 12,000 which which means 16,000 was invested at. 31% at 3.1% rather so my answers are x equal 12,000 and as well 16,000 together they give me my total amount of 28,000 and we're done let's look at example six example five ends the ch challenging portion of this lesson the most challenging thing about an application is generating the equation yourself when you generate the equation yourself then you always run the risk of solving an equation that could perhaps be wrong but when you're given the equation then the problem is much easier case in point the one we have now we're given the equation and when I do that the only thing I need to do is figure out what p p is and what f is the percent whatever it is is p fortunately we do not have to convert that to a decimal because it's giving us percent p and then the flow or F is uh what does that represent liters of air per second and for whatever reason that can only range from 10 to 75 so what flow must a hood range have to remove 70% of the contaminants well if I have an equation and it's got two unknowns either they'll give me the percent and I have to find F or they'll give me f and I have to find the percent in this case it should be clear that they've given us a percentage of 70 and I need to find F so that's what I do p is equal to 1.06 F + 7.1 8 if p is 70 then 70 is equal to 1.06 f plus 7.18 and now all I have to do is solve for f hopefully you see that we do that by subtracting 7.18 from both sides and what is that 60 2 82 I don't know 70 minus 7.18 yes 6282 and then what remains divide by 1.06 Final Answer 59 62.82% and now the very last problem the projected per capita Health Care expenditures in the US where Y is in dollars and X is years after 2000 are given by this linear equation great the equation has already been generated for us I only need to know what Y is and I need to know what xes what were the per capita Health Care expenditures in 2007 and when will the per capita um expenditures reach 8500 all right we've been asked to do two things let's first figure out what our X is and what our Y is y is dollars okay the expenditures the health care expenditures and X is years after 2000 so my X years after 2000 my y the healthc care expenditures in dollars okay now let's look at the first part of the question what were the per capita healthc care expenditures in 2007 it should be clear that for this one we are solving for y and we've been given the year Well if x represents years after 2000 and it's 2007 2007 is how many years after 2000 seven so in other words find y when X is 7 that is the first thing that we're being asked Y is equal to 343 x + 4512 in 2007 there are seven years after 2000 and so we simply find y when X is 7 all right let's plug that in in 343 * 7 4512 6,913 and we're done with the first part okay when will the per capita expenditures reach 8500 I I'm really hoping it's clear with this second part that they've given us the dollar amount they've given us Y and they're asking us to find the year X so it's the same equation okay the only difference this time is instead of them giving us X and US needing to find y they're giving us Y and we have to find X X Y is 8500 and that is equal to 343 X Plus 4512 well it should be clear that the first thing we need to do is subtract 4512 from both sides that gives us 3988 and that is equal to 343 * X and the only thing that remains is to divide both sides by 343 let's see what we get 11 now what happened here I believe oh no not this one I believe it's just 11.63 now the very last part of our strategies for solving applications was to make sure that not only does our answer makes sense and that it's reasonable but I answer the question that has been asked the question that was asked is when will the per capita expenditure reach 8500 you think the answer to when is 11.63 no what does X represent X represents look the years after 2000 so if x is 11.63 then the answer is in 2011 the latter half and with that we have concluded the example um we have incl we have concluded rather section 1.2 with that example let's look at the quiz the length of a rectangle is 2 in more than three times the width what are the dimensions if the perimeter is 28 in now I want to make this clear this quiz is optional if you complete it I will give you extra credit what that means is um since at at present we've only done one quiz if you've already got 100% on it then I'll Bank it and save it until you get lower than that if you got lower than 100% on your first quiz then I will replace it with this one you are to turn it in um in Moodle in the assignments tab under turn it in if you cannot figure out how to do that no worries you can email it to me for now and I will provide further classific um further clarification when we meet again take care everyone bye-bye now