Algebra 1 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 unknown0069

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 30165

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 steps0180

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 number0242

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 one1084

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 unknown1409

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 piece1425

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 tricky1703

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