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### Applications of Linear Equations

- When solving word problems we go through 6 steps to arrive at a solution
- Read the problem
- Assign a variable to the unknown quantities
- Write an equation using the connections in the problem
- Solve the equation
- State your answer to fit the context of the problem
- Check your solution to see if it is reasonable
- You may have to repeat some steps a few times when working through a problem.
- Look for key words in the problem that will help you determine how the equation should be written.
- Always check your solution to see if it is reasonable in the context of the word problem.

### Applications of Linear Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:10
- Applications of Linear Equations 0:43
- The Six-Step Method to Solving Word Problems
- Common Terms
- Example 1 5:03
- Example 2 9:40
- Example 3 13:48
- Example 4 17:58
- Example 5 23:28

### Algebra 1 Online Course

### Transcription: Applications of Linear Equations

*Welcome back to www.educator.com.*0000

*In this lesson we will look at applications of linear equations.*0002

*Think of those linear equations but more of a word problem sense.*0006

*Some of the things we will cover is the six step method that you can use to approach some of these word problems and to use that method.*0012

*We will have to look at how you can start translating these math things into actual equations.*0020

*We will also loot at very specific types of word problems that you might encounter such as when you just have some unknown quantities involving numbers.*0028

*Some of that involve some trigonometry and ones that might just involve some consecutive integers.*0036

*How should you approach some of these word problems?*0046

*In the six step method, we like to try and understand as much about the problem as we can.*0049

*That is why in the first step we are looking at reading the problem as much as possible and gathering up information and trying to understand it.*0055

*Once we think we have enough information it is a good idea to assign some sort of variable to the unknown.*0063

*In some of these problems, it might look like there is more than one unknown*0069

*and in cases like that you should assign a variable to the one that you know the least amount about.*0072

*From there we will be able to actually start writing equation from the information of the problem.*0080

*This is probably the most difficult step and sometimes you might get stuck on that one.*0084

*Once we have an equation when you go ahead and move on to solving that equation and see if we can actually get a solution.*0091

*We are not done even when we do get a solution it is a good idea to state the solution in the context of the actual problem.*0098

*That way we know if we get done we will say x = 3.*0104

*What exactly does x represent? Is it the number of miles driven?*0107

*Is it the number of beans in our basket? What exactly is the variable?*0111

*It is also important that we identify it early because it will make step five much easier.*0117

*When we are all done, we want to make sure that we check our solution to make sure that it is reasonable.*0125

*Especially with a lot of word problem, sometimes you might get a solution that just does not make sense in the context of the problem.*0131

*For example maybe I'm going through on solving something and I actually get time being -3 seconds.*0137

*It might be a problem because we can have negative seconds being time.*0143

*It is definitely a good idea to check your problem, check your solution and make sure that they make sense.*0147

*One thing I will note is that these steps often make it look like a nice linear process like you have simply moved through 1, 2, 3, 4, 5, 6.*0154

*In practice that may not always happen.*0163

*Do not be too surprised if you end up going through step 1, step 2, and you try to build that equation in step 3*0165

*and you realize you might not have all the information you need.*0171

*In cases like that, it is good to go right back to step 1 and see if we can gather up even more information.*0175

*When you are completely done with solving the problem, you will eventually move through all 6 of these steps*0180

*but you may end up bouncing back a few times if it looks like you do not have quite enough information.*0185

*If I’m taking a lot of word problems and being able to create equations then we want to be on the lookout for some keywords that are in those word problems.*0196

*We will use these keywords, these nice common terms to help us translate some of what we see in the problem into the actual formula.*0204

*Here are some terms that you should be familiar with.*0212

*When we are looking at addition, you want to look for words such as sum, together, total, more than or added to,*0215

*that are usually your clues that something will be added.*0224

*Think of a number and 3, that would say I want x + 3.*0227

*For subtraction we are looking for words like difference, minus, less than and decreased by,*0234

*all of those usually signify subtraction.*0239

*You have to be a little bit careful on the order of subtraction, for example you might see something like 3 less than a number*0242

*and that would say you have like x – 3 because that would be 3 less than that number x.*0250

*The order is important and sometimes it seems a little backwards.*0257

*For multiplication, look for product, times, twice or three times they might say something like that and percent of.*0261

*Those all indicate that something should be multiplied together.*0270

*Division has some good terms, you are looking for quotient, divided by, or ratio.*0276

*In all of our equations, we will need an equal sign there somewhere so we will be looking for equals is or will be.*0283

*That would be a good way to signify where the equal will go in the equation.*0292

*We will definitely come over our problems and look for some of these key terms so we can build our equation and solve from there.*0296

*Let us go ahead and jump into some of the examples and we will see what we have.*0305

*Let us first just start off by reading it and seeing what information we can dry out.*0309

*This one says the quotient of a number and 6 is added to twice the number and the result is 8 less than the number, find the number.*0314

*It looks like I see lots of things packaged up in here and just coming over.*0324

*I'm dealing with the quotient, remember that is division.*0330

*Added to means there are some addition.*0334

*Twice the number means we have some multiplication.*0337

*The result is there is our equal sign.*0340

*8 less than the number, this will represent subtraction.*0344

*There are lots of good pieces that are flying around in there.*0349

*I think the very thing I need to do now that I got some good information is.*0352

*Let us go ahead and identify our unknown.*0356

*If I’m going to be using an x, also that x is the number and that is the one we are looking for.*0360

*We will use that in our equation to package up all the rest of this information.*0371

*We have identified are variable, let us see if we can come over, and start putting it together.*0376

*I have the quotient of a number and 6.*0382

*The quotient of our number and 6 would be division or x ÷ 6.*0386

*That entire piece is added to twice the number, notice how I’m using multiplication.*0393

*The result is equals 8 less than the number, this is the one that seems a little backwards.*0406

*But we will take our number and subtract 8 so that we can get 8 less than our actual number.*0415

*We have read it over, identified our variable, we set up our equation, now we have to move through the actual solving process.*0420

*In this process we use a lot of our tools as before.*0428

*We try and clear out some of our fractions with a common denominator and get those x’s isolated onto one side.*0431

*With this one, I’m going to multiply everything through by 6.*0437

*This will definitely help us take care of that fraction that x ÷ 6.*0453

*We will use our distributive property, that will give us x + 12x = 6x - 48.*0461

*Now we do not have to deal with our fractions and we can just work on simplifying each side of the equation and getting our x’s alone.*0477

*I see some like terms on the left, 1x and 12 x = 13x, 6x – 48.*0484

*I simplified each side now let us go ahead and move the 6x to the other side.*0495

*It looks like we are almost done.*0514

*Finally, we will divide by 7 the x will be completely isolated.*0517

*It looks like our number, the one we are looking for, is a -48/7.*0532

*In terms of does this fit the actual context of the problem?*0539

*This one is less of a real word problem that is why I do not have to deal with time or distance.*0542

*It seems a perfect reason if that this is the number we are looking for.*0547

*We can always check the number to make sure that it works by taking it, and putting it back into the original.*0552

*If I want to take this and put it all the way back into all of the x here it should work out just fine.*0559

*That is the process in a nutshell to get your solution.*0568

*Let us look at some ones that are little bit more real world-ish and see how we can move through those.*0573

*This one says the perimeter of a rectangle is 16 times the width and length is 12 cm more than the width.*0583

*Find the length and width of the rectangle.*0590

*It seems like a pretty good problem I want to get as much information on this as I can.*0594

*I’m going to start off by trying to draw a picture of what we are dealing with here.*0599

*It sounds like we have some sort of rectangle and the perimeter of the rectangle is 16 times the width.*0604

*It may be my unknown here should be the width.*0613

*Let us mark that w is the width.*0618

*The length is 12 cm more than the width and I can mark out and then find the length and width of the rectangle.*0630

*I draw a little picture to give myself a sense of what is going on.*0645

*I have labeled my unknown w being the width and now I think we have enough information to start packaging this together into an equation.*0649

*Let us see what we can do.*0657

*The perimeter would be the sum of all the sides.*0659

*w + w that would take care of these two sides right here, + w + 12 + w +12 and that will take care of both of my lengths.*0664

*That entire thing is the perimeter right there and it is equal to 16 times the width.*0687

*It looks like a pretty good equation.*0696

*We just have to work on solving to make it all makes sense.*0698

*Let us see on the left side I have lots of w's and we can go ahead and combine all of them together into just 4w since all of those are like terms.*0703

*Then I have a 12 + 12 so we will say that is 24, all equal to 16w.*0715

*Simplify each side, let us go ahead and put our w's together, 24 = 12w.*0727

*One last step, divide both sides by 12 and looks like we get that 2=w.*0740

*It starts the work where we interpreted in the context of the problem.*0753

*Since we identify this earlier that w is the width I know that this represents the width of the rectangle 2.*0756

*Since our units in here are in centimeters and let us say that our width is 2 cm.*0764

*That is all we wrote we would actually be in a lot of trouble because we have not answered the entire problem.*0778

*In the entire problem we want to know the length and the width of this rectangle.*0785

*We only have the width so let us use this width to see if we can find the other part.*0789

*Since it says that the length is 12 cm more than the width then we can simply add 12 to this 2 and we will get the other.*0796

*Length is 14 cm.*0807

*We have answered both parts of this question I know that the width is 2 cm and the length is 14 cm.*0819

*Let us look at another one, this one says during a sporting match the US team won 6 more medals than Norway and the two countries won a total of 44 medals.*0831

*How many did each country win?*0840

*This is one of those examples where I said it looks like there is more than one variable in here.*0842

*After all, we have no idea how many medals that US won and I have no idea how many medals Norway won.*0847

*Let us see if we can pick apart and see which one we know the least about.*0853

*During a sporting match the US team won 6 more medals than Norway, I know a little bit about the US,*0856

*they won 6 more than Norway and the two countries won a total of 44.*0862

*It looks like the one I know the least amount about is actually Norway.*0867

*I have no idea how many medals they won.*0871

*Let us set that up as our variable so x is the number of medals won by Norway.*0872

*If the US team won 6 more medals than Norway, I could represent that using addition, x + 6 that would represent the number of medals that the US won.*0898

*I want to look at their total wins and it should be 44.*0913

*I will add Norway's medals to that should equal 44.*0918

*Let me highlight the parts of this equation so you can see exactly what I'm doing here.*0923

*This x right here is the number of medals from Norway and this over here is the number of medals from the US.*0927

*You can see I'm taking both of those quantities and put it together and look at the total number of medals from both one.*0938

*Now that we have our equation, again let us work to solve it.*0946

*My two like terms I got x and x that would give me a 2x then I can work to get my x’s all by themselves.*0950

*Let us go ahead and subtract 6 from both sides 2x = 38.*0964

*Now we will divide by 2, this will give us x = 19.*0972

*We have got to the problem and we figured out what x is but again interpret it in the context of the problem.*0993

*x = 19 but what is x?*1000

*Earlier we said that x is the number of medals won by Norway so I know that Norway won 19 medals.*1003

*Let us write that down.*1011

*We are not done yet, we still have to say how much each country won.*1025

*I also need to know how many the US won.*1030

*Since we know that the US won exactly 6 more than Norway and I can just take 6 and add it to the 19, the US won 25 medals.*1033

*I just have to double check this quickly, add these together and make sure we get 44.*1050

*19 + 25 = 44 total, it looks like we are doing okay.*1054

*Let us look at another example and see how well that one turns out.*1074

*In this example, we are going to look at some angles and a little bit of trigonometry.*1080

*It is okay if you do not know anything about trigonometry, I will give you all the background information you need for this one*1084

*You can see how you need to put it together.*1089

*This one says find the measure of an angle such that its complementary angle and its supplementary angle both added to be 174°.*1092

*After looking at these terms in here and saying wait a minute what is supplementary and what is complementary?*1101

*Let me tell you.*1107

*Two angles are said to be complementary if you can add them together and they add to be 90°.*1108

*90° is complementary.*1114

*Two angles are said to be supplementary if you can add those together and you get 180°.*1117

*180° is supplementary.*1122

*What we are looking to do is you have some unknown angle and it has a complement and a supplement and we are looking at those two to get that 174°.*1126

*Let us first identify our unknown in this problem and that would be our angle, x is the unknown angle.*1137

*We have to use this unknown angle in order to build what its complementary angle would be and what its supplementary angle would be.*1154

*That is going to be a little bit tricky, because we want to know what would you have to add to this unknown one in order to get an x?*1162

*In order to get 90 and since we do not know, how could we figured that out.*1168

*Let us do a quick example.*1174

*What if we knew that our angle was 70°, what we have to add to that in order to get 90?*1175

*It would not take that long to figure out, you should just take another 20° and sure enough you get 90°.*1183

*If you think about how you came up with that, you will be amazed that all you have to do is take 90 and subtract the angle that you knew about.*1189

*We will say that 90 - x that right there will be our complementary angle.*1199

*In a similar way you can also build your supplementary angle.*1216

*You can say that it will be 180° minus whatever angle you started with.*1220

*We have our complementary angle and we have our supplementary angle.*1239

*In the context of the problem you want to be able to look at these two added together so that would be 174°.*1242

*Let us just take both of these expressions here, put them together.*1249

*(90 – x) + (180 – x) there is our complementary and supplementary must be equal to 174°.*1253

*Let us work on combining our like terms and see what we can get.*1266

*90 + 180 = 270, -x, -x, -2x = 174.*1270

*We will subtract 270 from both sides, -2x = -96 and then let us go ahead and divide by -2, x = 48.*1286

*It looks like our unknown angle is 48°.*1314

*We can check to make sure that this is our angle by using it back into the context of the problem.*1332

*If this angle is 48 then what do I have to add in order to get 90°?*1338

*In other words, what would be its complementary angle?*1343

*If I take 48 and I add 42 that will give me 90, so 42 is the complementary angle.*1353

*Let us do the similar process, what would I have to add to 48 in order to get 180° or even if you take 180 - 48 to see what its supplementary angle would be.*1365

*Looks like 132.*1388

*If we take both of these and add them together, sure enough we get 174° like we should, so we know that our angle is 48°.*1392

*One last example and in this one we will deal with looking at a lot of different numbers but there is only one unknown*1409

*and we will look at it as being the one we know the least amount about.*1416

*This one says we have a piece of wire and it is 80 feet long.*1421

*We are going to cut this into 3 different pieces with the longest piece being 10 feet more than the middle sized piece*1425

*and a short piece being 5 feet less than the middle sized piece.*1431

*Find the length of the three pieces.*1434

*It looks like we do not know anything about the short, the middle or the long one, but we will write them all just using one variable.*1438

*I know a little bit about all of them put together since I know that my total is 80 feet long.*1447

*I'm going to come in three pieces and the longest piece is 10 feet more than the middle.*1454

*I know a lot about the long one.*1460

*The middle sized piece and the shorter piece is 5 feet less than the middle sized piece.*1463

*I know a little bit about the middle one, its 5 feet more than the shorter one since the shorter one was 5 feet less.*1471

*I think what I need to set up as my unknown and we will say that x is the length of the short piece.*1479

*I will try and describe all the other pieces using just that short length right there.*1499

*The short piece is 5 feet less than the middle piece.*1505

*If I take x and I add 5 to it that should give me my middle piece just fine.*1514

*The long one is 10 feet more than the middle, then I can start with the middle and add another 10 and this would represent the long piece.*1528

*I need to take my short piece, my middle piece, and my long piece to be able to put all those together and represent the 80 total feet of wire.*1543

*Let us set up this equation.*1552

*short piece = x + our middle piece x + 5 + our long piece x + 15 all of this should equally total of 80 feet.*1554

*We got our equation so let us work to solve it.*1572

*Adding together our like pieces, I have 3 x’s but I can put together the 5 + 15 = 20.*1575

*I have 3x + 20 = 80.*1588

*Let us subtract 20 from both sides, giving us 3x = 60.*1593

*We will divide both sides by 3 and we will see what x needs to be, x = 20.*1604

*It looks like our shorter piece of wire is going to be 20 feet long.*1613

*We also want to identify what all the other ones need to be.*1636

*We can use these smaller expressions over here to figure out what they need to be.*1639

*Since the shorter piece is 5 feet less than the middle piece, you can simply add 5 to this and get that middle.*1646

*The middle piece is 25 feet long.*1654

*Our long piece since it is 10 more than middle one, we will add 10 + 25 = 35 feet long.*1668

*Now I have the information of all three bits of wire.*1685

*If we add all of these up, we should get a total of 80.*1691

*20 + 25= 45 + 35 sure enough adds up to a total of 80 feet.*1695

*You can see that the process of interpreting these word problems can be a little tricky*1703

*But if you look for those keywords and try and hunt down your unknowns as much as possible, it can work out pretty well.*1708

*You can always look at the context of the problem to make sure your answers make sense.*1714

*Thanks for watching www.educator.com.*1719

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 6:54 PM

Post by Pauline Nunn on August 26, 2016

I cant understand why in the beginning of example one you divided by 6...

i understand that it says quotient and that means dividing, but if i saw the question saying "the quotient of a number ",AND" 6," I would have for sure thought they were asking you to have some number divided, plus 6, plus 2x

How did you understand what "the quotient" means in relation to the veriable here?

i mean, how did you know that "AND" meant that it should be the denominator of the quotient/fraction?

1 answer

Last reply by: Professor Eric Smith

Fri Aug 26, 2016 6:58 PM

Post by Kevin Zhang on July 17, 2016

Hey Eric,

I think you might have made a mistake on slide the common terms slide in the division section. Instead of ratio or, I think you meant or ratio.

Otherwise, I enjoy your lessons and wonderful job. They are really easy to read/ clear and organized.

8 answers

Last reply by: Professor Eric Smith

Mon Mar 30, 2015 12:39 PM

Post by Denise Bermudez on March 7, 2015

Hi!

I am actually very very coonfused by the fact that you chose the middle piece. I had chosen the shortest piece because I thought I knew nothing of it. Besides that I am also very puzzled as to why you add a certain number instead of subtracting it

Ex. middle= x+5 instead of x-5.

You also did this in a later lesson but my question wouldnt go through.

thanks in advance

1 answer

Last reply by: Professor Eric Smith

Fri Dec 26, 2014 10:49 AM

Post by Mohamed Adan on December 25, 2014

Hey Eric,

Enjoying your lessons here. Want to ask about example 5. I like attempting the examples before watching how you solve them, and with 5 I selected the middle piece to be my variable because I felt it was the one I knew least about. You selected the shortest piece. We both got the same answer. I wanted to ask if it's typical for a word problem to have more than one variable that could be worked with, of there's usually only one?