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Lecture Comments (8)

3 answers

Last reply by: Professor Selhorst-Jones
Tue Nov 4, 2014 11:20 AM

Post by Magesh Prasanna on September 3, 2014

Hello sir! The order of operations is obviously correct but what would be the formal proof for the order of operations? The logical deductions?

1 answer

Last reply by: Professor Selhorst-Jones
Fri Feb 21, 2014 9:24 AM

Post by Linda Volti on February 20, 2014

Another fantastic lecture! Thank you!

1 answer

Last reply by: Professor Selhorst-Jones
Sun Dec 29, 2013 12:00 PM

Post by Abdihakim Mohamed on December 29, 2013

Example 1 isn't the 4 distributive

Related Articles:

Variables, Equations, & Algebra

  • A variable is a placeholder for a number. It is a symbol that stands in for a number. There are generally two ways to use a variable:
    • The variable is allowed to vary. As its value changes, it will affect something else (the output of a function, a different variable, or some other thing).
    • The variable is a fixed value (or represents multiple possible fixed values) that we do not know (yet) or do not want to write out. Normally we can figure out the value by using information in the problem.
  • We normally use lowercase letters to denote variables, but occasionally we will use Greek letters or other symbols.
  • A constant is a fixed, unchanging number. Occasionally, we might use a symbol to refer to a constant (In such a case, we might refer to it as a variable, but we know that since it's a constant, the variable is fixed.).
  • A coefficient is a multiplicative factor applied to a variable.
  • An expression is a string of mathematical symbols that make sense used together. Often we will simplify an expression by converting it into something with the same value, but easier to understand (and usually shorter). For example, we might simplify the expression 7+1+2 into the equivalent 10.
  • An equation is a statement that two expressions have the same value. We show this with the equals sign:  =. For example, the equation
    2x+7 = 47
    says that the expression 2x+7 is equivalent (equal) to the expression 47. In other words, each side of the equation has the same value.
  • If we have an equation (or other kinds of relationships as well), we can do algebra. The idea of algebra is that since each side is equivalent to the other side, if we do the exact same operation to both sides, the results must also be equivalent. This idea makes sense, but it's critically important to remember you must do the exact same thing to both sides when doing algebra. If you do different things on each side, you no longer have an equation. This is a common mistake, so don't let it happen to you!
  • When you solve an equation, you are looking for what value(s) make(s) the equation true. Most often you will do this by isolating the variable on one side: whatever is then on the other side must be its value. You isolate the variable by doing algebra. Ask yourself, "What operation would help get this variable alone?", then apply that operation to both sides.
  • It's critical to remember the order of operations when simplifying expressions and doing algebra. Certain operations take precedence over others. In order, it goes
    1. Parentheses (things in parentheses go first),
    2. Exponents and Roots,
    3. Multiplication and Division,
    4. Addition and Subtraction.
  • Distribution allows multiplication to act over parentheses. The number multiplying the parentheses multiplies each term inside the parentheses:
    3(5 + k + 7) = 3·5 + 3k + 3·7.
    We can also use the distributive property in reverse to "pull out" a factor that appears in multiple terms:
    3x2 + 7x2 − 5x2 = (3 + 7 − 5) x2.
  • We can use information from one equation in another equation through substitution. If we know that two things are equal to each other, we can substitute one for the other.
    x = 2z + 3,     5y = x−2        ⇒     5y = (2z+3) − 2.
    When we substitute, we need to treat the replacement the exact same way we treated what was initially there. The best way to do this is to always put your substitution in parentheses.

Variables, Equations, & Algebra

Consider this expression:
    −47t + 8.
What term is the variable?
What term is the coefficient?
What terms are the constants (there are two of them)?
  • The variable is a placeholder for a number. It is a symbol that stands in for a number. Which symbol in the expression is standing in for a number?
  • A coefficient is a multiplicative factor on a variable. Since t is our variable, what number is multiplying it?
  • A constant is a fixed, unchanging number. What numbers show up in the expression? [Notice that t is not a constant, because we don't know whether or not it can change.]
The variable is t.
The coefficient is −47.
The constants are −47 and 8.
Explain, in your own words, why we can solve the equation
x+2 = 8
for x by subtracting 2 from each side.
  • To solve an equation for something means to get that something alone on one side of the equals sign. In this case, if we're solving for x, we want to get x alone on one side.
  • The idea of algebra is that if we do the exact same thing to both sides of an equation, we obtain another equation.
Answers may vary; We know that x+2 and 8 are equivalent because of the equals sign. If we do the exact same operation to each side (in this case, subtract 2), our "new" left and right sides will still be equal. Thus, if we subtract 2 from both sides, we will have x alone on the left and it will equal whatever is on the right. This gives us x=6.
It is this idea of doing the exact same thing to both sides that algebra is based upon.
Solve for x:
2x − 14 = −8
  • We want to isolate x on one side. We choose all of our steps based on that goal.
  • Begin by adding 14 to both sides (this cancels out the −14 on the left).
  • Divide the resulting equation by 2 on both sides (this cancels out the 2 in 2x on the left).
x = 3
Simplify (pay attention to the order of operations!):
3 ·(42 − 10)
  • The order of operations goes in the following order: parentheses, exponents and roots, multiplication and division, addition and subtraction.
  • (42 − 10) is in parentheses, so we need to calculate it first.
  • Inside of the parentheses, we have an exponent: 42, so that happens before the subtraction.
    (42 − 10) = (16 − 10) = (6)
  • Now that we know what is inside the parentheses, we can multiply 3 by that number.
Simplify (make sure to use distribution!):
5(2a − b + c) + 3a + 2b
  • Distribution allows multiplication to act over parentheses.
    3( x+ 7) = 3x + 3·7
    The thing multiplying outside the parentheses distributes to each term inside of the parentheses.
  • Begin the problem by distributing the 5 to each term inside of the parentheses.
    5(2a − b +c) = 5·2a + 5·(−b) + 5·c
  • Multiply out, then add the result to the rest of the expression
    (10a − 5b + 5c) + 3a + 2b
13a − 3b + 5c
Simplify (use distribution and the order of operations):
12p + 3
22 (p+q) − 3 ·2 p
+ 10q
  • The order of operations goes in the following order: parentheses, exponents and roots, multiplication and division, addition and subtraction.
  • Because we have parentheses nested inside of parentheses, those go first. However, we can't simplify it anymore since p and q are different variables.
  • Next, we need to distribute on to (p+q), but we first need to know what 22 is. 22 = 4, so we multiply that on to (p+q).
  • We continue with the order of operations and now multiply −3 and 2p to get −6p.
  • At this point, we have shown
    12p + 3
    22 (p+q) − 3 ·2 p
    + 10q = 12p + 3
    4p+4q − 6p
    + 10q
  • We continue to simplify inside the parentheses, finally getting (−2p + 4q), which we then multiply by the 3 in front.
  • We are now at the last step in the order of operations: adding everything together.
6p + 22q
Solve for k:
3k + 18 = 3(5−2)
  • Start by simplifying the equation (pay attention to the order of operations):
    3k + 18 = 9
  • We want to isolate k on one side. We choose the rest of our steps based on that goal.
  • Get rid of the 18 on the left side by subtracting it from both sides.
  • Get rid of the 3 on the 3k by dividing both sides by 3.
k = −3
Solve for x+2y:
5x + 10 y −35 = 25
  • When solving for something, we are isolating it on one side. In this case, we aren't isolating just one variable, we're isolating the whole expression x+2y. We choose the rest of our steps based on that goal.
  • Get rid of the −35 on the left side by adding 35 to both sides. This gives
    5x + 10 y = 60.
  • Notice that 5x + 10y is a multiple of x+2y. Both the terms have just been multiplied by 5. Thus, we can divide both sides of the equation by 5 to reach our goal.
x+2y = 12
Use substitution to solve for n:
n = g2               g = −3 ·5 + 17
  • To find n, we need to substitute the value of g in to n=g2.
  • We have two options. First, we could just substitute in g immediately to get
    n = (−3 ·5 + 17)2,
    then work out the answer. Alternatively, we could first work out the value of g, then plug it in (substitute it in). This seems slightly easier, so we will do that [although the other method is perfectly fine].
  • Simplifying g, we get g = 2.
  • Plug g=2 into n = g2 to obtain
    n = (2)2.
Solve for t in terms of s:
r = 3s               q = −s+3               3t − 11s = 5q + 2r2
  • The phrase "solve for t in terms of s" means that we need to isolate t on one side (`solve for t') and the only other variable that is allowed to appear in the final equation is s (`in terms of s').
  • This means we need to eliminate the variables q and r from the equation. We can do this by using substitution. Since we can see that q and r are in terms of s in the other equations, we can replace them in the equation containing t.
  • When we substitute in, it is crucial to wrap our substitution with parentheses or things will go very wrong. Correctly doing so, we get
    3t − 11s = 5(−s+3) + 2(3s)2.
  • Notice that (3s)2 = 9s2. Simplifying the right side of the equation, we have
    3t − 11s = −5s +15 + 18s2.
  • Now we isolate for t: add 11s to both sides, then divide both sides by 3.
t = 2s + 5 + 6s2.
Later on when we learn about polynomials, we'll learn that we often order terms by the exponent on the variable. While the above is correct, some teachers might like to see that now: t = 6s2 + 2s + 5. Notice how that is equivalent to the above, just in a different order.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.


Variables, Equations, & Algebra

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
  • What is a Variable? 0:05
    • A Variable is a Placeholder for a Number
    • Affects the Output of a Function or a Dependent Variable
  • Naming Variables 1:51
    • Useful to Use Symbols
  • What is a Constant? 4:14
    • A Constant is a Fixed, Unchanging Number
    • We Might Refer to a Symbol Representing a Number as a Constant
  • What is a Coefficient? 5:33
    • A Coefficient is a Multiplicative Factor on a Variable
    • Not All Coefficients are Constants
  • Expressions and Equations 6:42
    • An Expression is a String of Mathematical Symbols That Make Sense Used Together
    • An Equation is a Statement That Two Expression Have the Same Value
  • The Idea of Algebra 8:51
    • Equality
    • If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
    • Always Do The Exact Same Thing to Both Sides
  • Solving Equations 13:23
    • When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
    • Look For What Values Makes the Equation True
    • Isolate the Variable by Doing Algebra
  • Order of Operations 16:02
    • Why Certain Operations are Grouped
    • When You Don't Have to Worry About Order
  • Distributive Property 18:15
    • It Allows Multiplication to Act Over Addition in Parentheses
    • We Can Use the Distributive Property in Reverse to Combine Like Terms
  • Substitution 20:03
    • Use Information From One Equation in Another Equation
    • Put Your Substitution in Parentheses
  • Example 1 23:17
  • Example 2 25:49
  • Example 3 28:11
  • Example 4 30:02

Transcription: Variables, Equations, & Algebra

Hi--welcome back to

Today we are going to talk about variables, equations, and algebra.0002

What is a variable? We talk about them all the time: we want to think of a variable as just being a placeholder.0006

It is a placeholder for a number; it is a symbol that stands in for something that can come in later; it is standing in for a number.0011

Sometimes the variable will be able to vary; it is going to be able to change, depending on what we want to do.0019

And as the value of the variable changes, it will affect something else.0025

It might affect the output of the function; it might affect some other dependent variable,0028

if we see something like y = 3x, where you change x--we make it the independent variable.0033

So, we put in different things for x, and it causes our dependent variable, y, to change, varying on what we put in for x, or something else.0039

That is one way of looking at a variable: it is something that is allowed to vary, and it causes other things to shift around as it changes.0046

Other times, we are just using a variable as a fixed value we don't know yet.0053

Sometimes, it might even be multiple possible fixed values; it could be fixed values or a fixed value.0057

But the point is that it is something that we just don't know yet; it is a placeholder for something that we want to find out more about.0062

So, normally, we are going to be able to figure out what it is, based on the information given to us in the problem.0070

Otherwise, it is probably not going to be a very good problem, if we can't actually solve for what the variable is.0075

So, we will almost always have enough information to figure out what this variable is.0080

That is the other possibility: a variable is something that we just don't know yet.0085

It is a number that has been given a name, because we are trying to figure out more information about it.0089

It is like if a detective is trying to find out who committed a crime; they might talk about the perpetrator, and they might find facts out0093

about the perpetrator, until they have enough information to be able to figure out who the perpetrator actually is.0100

"Perpetrator" is just a placeholder for some other person, until they figure out who that person is.0106

Great; we can name variables any symbol that we want.0111

Normally, we are going to use lowercase letters to denote variables; but occasionally, we will use Greek letters or other symbols.0115

When we are working on word problems, we are going to choose our symbols...we want to choose the letter,0121

or maybe other symbol that we use, based on something that helps us remember what it is representing in this word problem.0125

What am I using this variable to get across?0132

There are a lot of them that we regularly use; and so, we will get an idea of what they are; here we go!0135

Any symbol could potentially be used for any meaning at all.0142

We could make a smiley face; and I sometimes do use a smiley face to represent a number.0145

But smiley face is a little bit harder to draw than x, so we tend to use letters that we are used to drawing.0150

Anything could potentially be used for anything else; but here is a list of common symbols used,0156

and what the meaning we normally associate with them is.0161

Occasionally, we will have different meanings associated with them, depending on the problem.0163

We might use y to talk about the number of yaks that there are at a farm;0167

but generally, we are going to use them as we see right here--all of this right here.0171

So, x is our most common one, probably our favorite variable of all.0176

We use it for general use, when we are talking about horizontal location or distance.0180

y is vertical location or distance; t normally stands in for time; n stands in for a quantity of some stuff.0185

θ--this is a Greek letter, theta; when we encounter Greek letters, I will talk about them a little bit more, but mainly, it is just going to be θ.0192

We draw θ by hand; you just make something sort of like you are drawing a zero or an O,0199

and then you just draw a line straight across the middle of it; that is θ.0205

r is radius; A is area; V is volume; and we often use a, b, c, and k to represent fixed, unchanging values--0209

values that are not going to vary and change into other things--things where we know0221

that they are going to just stay the same, but we don't know what they are yet.0224

Or we might decide on what they are later on.0228

Anyway, this gives us a general idea of what the normal stable variables we constantly encounter are.0231

Now, you might use x for something totally different than what we have here.0238

You might use r for something totally different; you are not stuck to just using this.0241

But we are going to see them in a lot of problems, and we want what we do to make sense to other people.0244

So, it is good to go along with these conventions, usually.0249

All right, what is a constant? A constant is a fixed, unchanging number.0253

It is a value that does not turn into another value.0258

So, we can have variables become different values; we might plug in x = 3, and then plug in x = 5, and then plug in x = 7.0261

But a constant only has one thing; it just stays the same.0269

So, any time we see a number, like 3 or 5.7 or -82 or anything that is just a number, it is a constant, because numbers do not change.0272

After all, we don't have to worry about 3 suddenly turning into 4.0281

It is just 3; it is going to be 3 today; it is going to be 3 tomorrow; it is going to be 3 forever.0284

3 doesn't suddenly jump around and become a new number.0288

Occasionally, we might refer to a symbol that is representing a number as a constant.0292

We might say a is a constant in this problem.0295

We might not know what value that symbol represents; but we know it cannot change--a constant is something that cannot change.0298

And other times, we might even refer to a symbol as a variable, and just know that that variable is fixed, that it is a constant variable.0305

It seems kind of like a contradiction in terms, but remember: we are using "variable" more for the idea of placeholder.0311

And while sometimes it varies, sometimes it can also just be a placeholder in general.0316

A constant is something that isn't going to move around; it is one number, and one number only.0321

It doesn't matter if it is a symbol, or if it is actually a number; but the idea is that it is something that is not going to change.0326

A coefficient is a multiplicative factor on a variable.0333

So, anything that has some number multiplying in front of it, and it is a variable, like 3 times x...its coefficient is 3.0336

Normally, it is just going to end up being a number; but occasionally, it is also going to involve other variables.0346

So, not all coefficients are constants, and not all constants are coefficients.0352

For example, if we have n times x, plus 7, we have n as the coefficient of x, because it is multiplied against x.0356

But 7 is not a coefficient, because it is not multiplying against any variable.0364

7 is a constant, though, because it is just a fixed number.0369

So, n is a coefficient, but it might not be a constant--it might be allowed to vary.0372

But it isn't going to be...n is probably not a constant, but it is definitely a coefficient.0377

And 7 is not a coefficient, but it is definitely a constant.0384

And we could even look at x as being a coefficient on n: we can look at it from n's point of view, or look at it from x's point of view.0387

So, a coefficient is a multiplicative factor; a constant is just something that doesn't change.0397

An expression is a string of mathematical symbols that makes sense.0403

What do I mean by "makes sense"? Well, you can put together a string of words in English that makes no sense,0407

like tree sound running carpet; that didn't make any sense, right?--tree sound running carpet--that was meaningless.0414

But it was a bunch of words; to be an expression in math means that you have to make sense.0423

So, to be an expression in English (passing this idea along as a metaphor) would mean that it has to make sense as a sentence.0428

A string of mathematical symbols that make sense together: 2 times 3 minus 5 could be an expression.0435

But (((((( times divide minus 4 times plus (...that doesn't make any sense; that is just a bunch of things that have been put down on paper.0444

They have just been written down, but they don't actually mean anything.0464

So, an expression has to make sense; that is one of the basic ideas behind it.0468

Often, we will need to simplify an expression by converting it into something0472

that has the exact same value, but is easier to understand, and often is just shorter.0475

For example, we might simplify 7 + 1 + 2 into the equivalent 10: 7 + 1 is 8, and 8 + 2 is 10; so 7 + 1 + 2 has the same value as 10.0480

They are both different expressions; they are different expressions, but they have the same value, so we can convert one to the other.0492

We can simplify it if we want to.0497

An equation is a statement that two expressions have the same value.0500

We show it with an equals sign: what is on the left side of the equals sign and what is on the right side of the equals sign--0504

we know that those two things are the same--they have the same value.0511

Each side of the equation might look very different: 3x + 82 looks very different that 110/2.0515

But that equals sign is telling us that what is on the left is the same as what is on the right.0522

It guarantees equality between the two sides.0526

Algebra...for being able to do algebra, we need to have some sort of relationship between two or more expressions.0532

In this course, our relationship is almost always going to end up being equality.0539

It is going to be based on having an equals sign between two expressions.0543

The two expressions will be equal to each other, and that gives us an equation to work with and allows us to do algebra and do some things.0547

We could potentially have a relationship that is not based on equality.0552

We could have an inequality, where one side is less than another side, or one side is greater that another side.0555

Or we could have another relationship that is different than either of those.0561

But for this course, we are going to almost entirely see equality; and that is going to make up pretty much all of the relationships we ever see in math.0565

They will be based around knowing equality between two things.0572

So, the two expressions will be equal to each other; and this gives us a starting point to work from.0576

The key idea behind math...not behind all math, but behind simple; it is intuitive; and it is incredibly important.0582

If two things are the same, equal, then we can do the exact same operation to both things; and the results will have to be the same.0590

Let's look at it like this: if we have a carrot here, and then we have another carrot that is exactly like that first carrot,0600

so they are a perfect copy of one another...we have two carrots, and then we come along and pick up a knife,0610

and we cut up this carrot with the knife, and we cut up this carrot with the same knife--0617

the exact same knife being used on both of them--and we cut up both of the carrots in the exact same way;0621

we cut 1-inch sections, exactly the same, on both of them; we are going to end up having chopped carrots0627

from the first carrot, and from the second carrot; but we know, because we did the exact same way of cutting them up,0634

and we started with the exact same carrot--our chopped carrot piles will be exactly equal to one another.0645

Since we start with the same carrot, and then we do the exact same kind of thing to both of the carrots,0652

we will end up having the exact same pile of chopped carrot at the end.0659

Now, compare that to if we had a third carrot that was exactly the same as its other two carrot "brothers,"0663

but instead of using a knife on it, we decide to shove it into a blender.0669

We put it in a blender, and we run it for a minute.0677

Out of that blender, we are going to get a pile of carrot mush; we are going to have a carrot mush pile.0680

And that carrot mush pile is going to be nothing like those chopped carrots.0687

It doesn't matter that we just started with equality; we also have to do the same thing.0692

Starting with equality is important; but if we don't do the same thing to both objects--0697

we don't do the same thing to both sides of our equation--we end up with totally different things.0701

We no longer have that relationship of equality that we really want to be operating on.0706

If you shove the carrot into a blender, you are going to have something totally different than if you had chopped it up.0711

If we do the exact same chopping to the two carrots, we end up getting the same pile of carrots.0715

But if we do a totally different thing, like shove it into a blender, we have something totally different at the end.0722

We have this pile of carrot mush; that is nothing like what we have from the other two.0727

The idea here is that we have to have the same operation be applied to both.0732

Doing algebra is based around this idea of doing the same thing to both sides.0737

Now, of course, you have seen this idea before: but it is absolutely critical to remember.0742

You have to remember this fact: always do the exact same thing to both sides.0747

If you don't do the exact same thing to both sides, you are not doing algebra anymore; you are just making fantasies up.0755

You have to do the same thing: if you add 7 to one side, you have to add 7 to the right side.0761

If you square the left side, you have to square the right side.0766

If you say higgledy-piggledy to the left side, you have to say higgledy-piggledy to the right side.0768

The huge quantity of mistakes that students make are because they forgot to do the operation on both sides.0772

They used it only on one side, or they used slightly different operations on the two sides.0779

If you end up doing this, you are going to end up making mistakes; don't let this happen to you.0786

Pay close attention when you are doing algebra--make sure you are doing the exact same thing to both sides.0791

You have to follow all of the rules on both sides; otherwise, we are just making things up--we are no longer following algebra.0796

When you are asked to solve an equation, you are being asked to solve for something.0804

This usually means solving the equation for whatever variable is in it.0809

If more than one variable is present, you will be told which variable to solve for.0813

What does solving an equation mean? It means you are looking for the things that make the equation true.0817

You are told that this side equals this side; the stuff on the left equals the stuff on the right.0824

But they both have variables in them, or one side has variables in it, or one side has just one variable in it.0831

But the point is that, depending on what that variable is, or depending on what those variables are, that equation might no longer be true.0836

So, what you need to do is make sure that this is true.0843

You were told that it equals one another; so you have to figure out what variable, what value for my variable,0848

or what values for my variables, will make this equation continue to be true.0854

I was told it was true from the beginning; so I have to make sure that it stays true.0859

Most often, you will be able to figure out what the values are that make something true by isolating a variable (or variables) on one side.0864

You will isolate the variable on one side, and then whatever is on the other side must be the value of that variable.0871

How are you going to do that in general?0877

Normally, you are going to isolate the variable by doing algebra.0879

You will ask yourself what operation would help get this variable alone.0881

What would I have to do to this side to be able to get this variable on its own?0886

Then, you do that operation to both sides; you continue to apply these operations, asking yourself,0890

one time after another, "What could I do to get this variable alone?"0895

You keep asking yourself, "What could I do?"; you keep doing operations to both sides.0899

And then, you keep doing this until, eventually, the variable is alone on one side, and you have solved it.0904

You will get something in the form like x = ... of numbers--so you will know that x is equal to this stuff right here; you will have solved it.0908

Now, keep in mind: sometimes you will not solve something by directly doing algebra.0917

Algebra will probably be involved, but you might actually be doing something a little bit more creative.0921

For example, we will see stuff like this when we work on polynomials.0926

We will see cases where we are not just doing algebra; we are also trying to figure out some other stuff and think on a slightly higher level.0929

But the key idea is that we are figuring out what makes this equation true.0936

What are all the possible ways to make this equation be true?0941

That is the real heart behind solving an equation.0945

It just so happens that it is very often a good way to solve it by doing algebra and getting the variable alone,0949

because once you get the variable alone and on one side, that tells you what value would make that original equation true.0954

Order of operations: it is critical to remember the order of operations.0963

We have known about this for a long time, but it still matters today; and it is going to matter for as long as you are doing math.0966

Certain operations take precedence over others.0971

In order, it goes: parentheses (things in parentheses go first), then exponents and roots, multiplication and division, addition and subtraction.0974

Always pay attention to the order of operations.0982

If you forget to do the order of operations, and you do it in a different order, disaster will befall your arithmetic.0985

So, always make sure you are working based on this idea of the order of operations.0989

Also, I just want to point out something: exponents and roots are two sides of the same thing.0993

x2 reverses square root: x2, √x...if you take something,0998

and you square it, and then you take its square root, they reverse one another.1003

Multiplication and division reverse one another: if we multiply by 3, and then divide by 3, it reverses.1006

Addition and subtraction reverse one another: if we add 5, and then we subtract 5, they reverse one another.1013

So, exponents and roots--the reason why they go at the same time is because they are really two sides of the same thing.1019

They have some similar idea going on behind them.1024

We will talk about that more when we get into exponents more, later in the course.1027

Multiplication and division: they go together at the same time, because they are two sides of the same thing; they can reverse one another.1030

Addition and subtraction go together at the same time, because they are working together; they are, once again, things that can reverse one another.1035

So, that is why we have these things paired together.1041

Parentheses, exponents/roots, multiplication/division, addition/subtraction: always make sure1043

that you are working in that order, or at least that whatever you are doing goes along with that order.1047

Sometimes, you might be able to do things where you don't have to follow this order absolutely precisely,1052

because you might see something like 3 times 2, plus (7 - 5).1056

Well, because there is this plus sign in the middle, we know that we can actually do what is on the left side1063

and what is on the right side simultaneously, because they will never talk to each other1068

until both orders of operations have completely gone through on their two sides.1072

So, we can just skip right to 6 + 2 = 8; we don't have to do everything there.1076

But if you are not quite sure--if you are not really capable with the order of operations,1082

so that you can see this sort of thing right away, always go with the order of operations very carefully, very explicitly.1085

In the worst case, it will just take a little bit longer, but at least you will not make a mistake.1091

Distributive property: we do not want to forget about the distributive property.1096

It allows multiplication to act over addition when it is inside of parentheses.1099

So, if we have 3 times (5 + k + 7), then that is equal to 3 times the first one, plus 3 times the second one, plus 3 times the third one (7)...1103

so 3 times 5, plus 3 times k, plus 3 times 7; that is the distributive property.1115

Always make sure you distribute to all of the terms that are inside of the parentheses; we have to distribute to everything inside of the parentheses.1120

I see lots of students see something like this, and they say, "Oh, 3 times 5, plus k, plus 7!"1127

No, no, no, no, no! You have to do everything inside of the parentheses; otherwise, you are not distributing.1132

So, make sure that you are always distributing to everything in there--everything, when you are multiplying in there.1138

All right, we can also use the distributive property in reverse, so to speak; we can go backwards, in a way.1144

This idea is what allows us to combine like terms.1150

For example, if we have 3x2 + 7x2 - 5x2, well, we have x2 here,1153

x2 here, and x2 here; so we can just pick them all up,1157

and we can shove them in, because they are all multiplying.1162

We pick them all up; and it is times x2; so we have (3 + 7 - 5) times x2,1165

because if we did the distributive property again, we would get what we started with; so it must be the same thing.1170

Now, 3 + 7 - 5--well, that just comes out to be 5: 3 + 7 is 10, minus 5 is 5; we get 5x2.1174

And that is what we are using to allow us to combine like terms.1182

We are sort of pulling out the like term, doing the things, and then putting it back in.1185

At this point, we have gotten so used to doing it that we don't have to explicitly do this.1190

But for some problems, it will end up being a really useful thing to notice.1193

So, it is important to see that we can occasionally use the distributive property in reverse; sometimes it will help us see what is going on.1197

Substitution: this is a really important idea in math.1204

We can use information from one equation in another equation through substitution.1207

If we know that two things are equal to each other, we can substitute one for the other.1212

For example, if we know that x is equal to 2z + 3, and we also have this equation that 5y = x - 2,1215

well, we can say, "Oh, look, right here I have an x, and I also know that x is the same thing as saying 2z + 3."1223

So, we take this information, and we plug it in for x.1231

That is what gets us (2z + 3); we will replace that x; so we have 5y is also equal to (2z + 3) - 2.1235

When we substitute, we need to treat the replacement the exact same way we treated what was initially there.1244

The best way to do this is by putting your substitution in parentheses.1254

Notice how I took 2z + 3, and I put it in parentheses up here, even though right here, it didn't start in parentheses.1257

That is because I was substituting in for x; so I want to make sure 2z + 3 is treated the exact same way that x was treated.1263

So, I have to put it in parentheses to make sure that it gets treated the exact same way that x got done.1269

The best way to do this is always to just put your substitution in parentheses.1275

It won't always be necessary: for example, on that 5y = 2z + 3, we didn't actually have to put it in parentheses there.1279

But it will never cause us to make a mistake; it is never going to hurt us.1285

(2z + 3) is just the same thing as 2z + 3, in this case right up here.1290

And in other cases (like this one that we are about to talk about), it is absolutely necessary; otherwise we will make bad mistakes.1294

Consider this really common mistake: if we know that a is equal b + 2, and we know that c is equal to a2,1301

then we can say, "Oh, a is right here; a is right here; I will take b + 2, and I will substitute it in for a."1308

Lots of students will say, "Oh, well, it is a2, so it must be b2 + 2."1315

No, that is not the same thing: we need a to be all of what it is.1319

a is all of (b + 2), not just the b part; and c is similarly not going to be equal to b + 22.1323

This right here is not working, because it has to be over this and this; everything needs to be put together.1331

b2 is not going to work here, as well.1338

The thing that we have to do is: we have to have it in parentheses.1340

The parentheses cause us to treat that a the same way that we are going to treat (b + 2).1343

a2...since a is equal to b + 2, all of a has to be squared; all of that (b + 2) has to be squared.1349

And the way that we get all of it is by putting it in parentheses.1356

So, whenever you are substituting something in, make sure that whatever is getting substituted in gets plugged inside of parentheses.1360

Otherwise, lots of bad mistakes can happen.1369

Sometimes, when you see the problem, you will be able to say, "Oh, I don't actually have to plug it in in parentheses,"1372

at which point, yes, you might be right; sometimes it will make it a little bit faster.1376

But really, it is a possible risk that you are taking for just putting down ().1380

It is not that much effort to put down parentheses, and it is going to save you so many times.1386

So, I really recommend that you put all of your substitutions, any time you are substituting something in, in parentheses.1390

Let's do some examples: we want to simplify the following: 2 times 32 + 4((5 + 7)2 - 27).1397

Well, we have parentheses inside of parentheses; so first, let's work on the thing inside of the parentheses.1407

And then, inside of that, we have even more parentheses.1413

So, first we do 5 + 7; we bring everything down--each new horizontal line is a copy of what was above it, but just put in new ways of talking about it.1415

4 times the quantity...well, what does 5 + 7 become? 5 + 7 becomes 12, so 12 times 2 minus 27.1425

Now, we keep doing this inside of these parentheses: first 2 times 32, plus 4...12 times 2 becomes 24, minus 27.1434

2 times 32...still working inside of these parentheses...24 - 27 becomes -3.1445

Now, we have 4 times -3, so now there is no longer anything happening inside of the parentheses.1454

So, what is next on the order of operations? Parentheses, then exponents and roots.1458

So, 2 times 32...32 is 9, plus 4(-3).1462

Next, we have multiplication: 18 + 4(-3)...-12; finally, we are down to addition and subtraction: 18 + -12 becomes 6; our answer is 6.1469

One thing I would like to point out is: if we are really good at math, we might have been able to say,1482

"Oh, look, there is a plus sign between these two sides, so these two sides aren't going to be able to talk to each other1485

until they have done everything they have to do on their own two sides."1491

So, we could have gone right down to saying 2 times 32...that is the same thing as 2 times 9, which is the same thing as 18.1494

And then, we would have kept doing our stuff on the right side, but we could have been simultaneously doing everything on the left side,1502

because they are not able to talk to each other, because they have plus signs between them and everything else.1508

That is a more advanced trick, and you are probably at the point where you can start seeing this sort of thing.1514

But if you have difficulty with the order of operations, you end up making mistakes like this sometimes.1518

Be careful and go through it really carefully, and make sure you have that stuff completely understood.1522

You need that foundation before math is going to be able to work.1526

It is the grammar of math; it is like knowing the grammar of English.1528

If you don't put words in the right order, it is just nonsense.1532

If you don't follow the operations in the right order, it is just nonsense; we are not able to speak the same language1535

as everyone else is speaking in math, and what everyone else is expecting us to be able to do1539

when we are working on problems or solving things...or engineering bridges...whatever we are going to do with math.1544

All right, Example 2: Use the distributive property to simplify 5(x + x2) + 3(x + y) - 7(x2 + x + y).1549

So, 5 times (x + x2) becomes 5x + 5x2.1560

Plus 3(x + y) becomes + 3x + 3y; minus 7...oh, here is something we have to be careful about.1567

It is not just going to be minus 7x2, but minus 7 is the entire thing.1576

So, it is that -7 that gets distributed; it is easier to see this as +, and then a -7.1581

+ -7: -7x2 + -7x + -7y--we have to make sure we distribute that negative, as well.1587

We see a minus, but it means that the "negative-ness" has to be distributed to everything inside of there.1600

Now, at this point, so we can see things a little bit more easily, let's move things together.1605

5x2...and here is a little trick: if you are not sure...we have 1, 2, 3, 4, 5, 6, 71610

different terms here--lots of different terms here to have to work with.1616

We can say, "Let's mark off each one; we will make a little tick mark after we write it on the next line, so we don't get confused,1619

accidentally use the same thing twice, or not even use it once."1626

So, 5x2 +...what is another thing involving x2? -7x2.1629

Plus...what comes next? It looks like we can work on the x's next: 5x (tick there) + 3x (tick there) - 7x (plus -7x);1637

and then finally, we have the y's: + 3y + -7y.1652

Those tick marks just help us keep track of what we are doing.1658

They are not necessary, but it makes it easier to follow, so we don't accidentally make any mistakes.1660

5x2 + -7x2...those will combine to become -2x2.1665

5x + 3x + -7x...we have 8x - 7x; we have 1x, which we just write as x.1671

And 3y + -7y becomes -4y; -4y we can also just write as minus 4y; and there is our answer.1681

Third example: we want to solve for x, so the first thing we do is ask ourselves, "How can I get x by itself?"1692

How can I get it isolated on one side, where it is just the variable, and only one of the variable, and nothing else there?1699

So, we say, "Well, it is inside of a fraction; we want it to be on top, and we want it to be the only thing there."1705

So, we are going to have to somehow change this fraction; how do we change a fraction?1713

Well, multiply by x + 3, and that will destroy the denominator.1716

Great--but if we multiply by x + 3, then this 2 is going to get hit, and this 3 is going to get hit, by the x + 3.1721

We have to hit everything on both sides, so the 3 will get hit by x + 3; the fraction will get hit by x + 3; and the 2 will get hit by x + 3.1728

So, the first thing we want to do is have some way of being able to have it operate on fewer things.1735

Let's try to get it to operate just on the fraction, at least on one side.1739

What we will do is start by subtracting 2 from both sides; that will make it easier to have a simple time with that x + 3.1743

We won't have anything else getting in the way.1750

That gets us 1 = 5/(x + 3).1753

Now, we can multiply by (x + 3); and while we will still have to multiply the 1 (we have to multiply both sides),1759

we will have at least a little less stuff in the way.1765

We multiply by (x + 3) over here, and we multiply by (x + 3) over here; so (x + 3) times (x + 3) on the bottom...they cancel each other out.1768

(x + 3) times 1...that is just going to become (x + 3).1780

Since we canceled out the (x + 3) on the bottom, we have 5 here.1785

Now, we ask ourselves, "How can I get that x alone?"1788

Oh, it is not too hard from here: we just subtract 3 from both sides: minus 3, minus 3; we get x = 2, and there is our answer; great.1791

Example 4--the final example: this one is a little bit tough, but we can totally understand what is going on.1802

x = 2z; y = z + 4; we want to solve for a in terms of z.1808

So, we have a in this equation down here; and we have x2, and we have y, and we have x.1814

So, z doesn't currently show up in this equation; we want to solve for a in terms of z.1821

What that means is that we want to get a = .... with z's...z's are going to be inside of that stuff on the right side.1825

And we are going to have a by itself: that is what "solve for a in terms of z" means.1834

a equals stuff involving z; it may be multiple z's; it may be just one z; but it is going to be a = [stuff involving z].1839

But notice: it is not going to involve x; it is not going to involve y.1848

We are told to solve in terms of z, so it is going to be only in terms of z and other actual numbers--that is, constant numbers.1851

So, if we know that x = 2z and y = z + 4, we need to get that z stuff to show up here, and we need to get rid of the y and get rid of the x.1859

So, we will use substitution: x = 2z and y = z + 4.1866

So, right here, we have a y; here we have an x; here we have an x.1877

So, let's do substitution: we have the left side--it will still be the same: 2a - 26 =.1883

What comes in for x? 2z comes in; so (2z)2 + 4 - 2(3...what goes inside for y? (z + 4)), minus...1889

what goes inside for x? (2z); close up that.1908

What we have is our original equation, but we have now gotten rid of x and gotten rid of y, and we only have z's and a's in here.1913

Now, we are able to solve for a in terms of z.1920

So, let's simplify what we have on the right side.1923

2a - 26 =...we have this plus sign in the middle, so we can work out what is on the left and what is on the right simultaneously.1926

We don't have to worry about them interfering with each other, even though they don't show up at the same time in the order of operations.1935

The only time they will be able to talk to each other is when we get all the way down to +.1939

So, we can have stuff on the left and stuff on the right work simultaneously to make it a little bit faster.1943

(2z)2...we square the 2; we square the z; so 22 and z2,1948

plus 4 minus 2...we go inside...3 times z, plus 3 times 4, minus 2z.1957

2a - 26 =...square 2; we get 4; square z; we don't know what z is, so it just stays as z2;1971

plus 4 minus 2 times (3z + 12 now - 2z)...keep simplifying...2a - 26 = 4z2 + 4 - 2(3z - 2z becomes just 1z + 12).1980

We can now distribute this -2: 2a - 26 = 4z2 + 4...we distribute the -2, so remember, it is plus a -2; so we get + -2z + -24.2004

Now, we are in a position to be able to keep simplifying the right side to its most fundamental level.2023

It equals 4z2...we don't have any other z2s, so it is just 4z2.2029

Plus 4...well, let's put our constants in there again; so we will go to + -2z, so - 2z, and 4 + -24 becomes - 20.2033

At this point, we can now do our algebra.2043

We will add 26 to both sides, and we will get 2a = 4z2 - 2z + 26, so plus...oops, I accidentally wrote what I was saying...2045

minus 2z still; add 26; -20 plus 26 becomes just 6; divide both sides by 2: a = (4z2 - 2z + 6), all over 2.2060

And we can simplify that: 4z2 becomes 2z2, minus 2z; that becomes -z; plus 6--that becomes + 3.2077

So, it is a = 2z2 - z + 3; a equals stuff-just-involving-z.2086

Solve for a in terms of z; great.2095

I really want to point out that the reason we were able to get that right is because we put parentheses when we substituted in.2097

If we hadn't done that, we would not have had our square go onto both the 2 and the z.2105

We would not have had our 3 distribute to both the z and the + 4; we wouldn't have had our subtraction...2110

well, our subtraction actually still would have subtracted 2z.2115

But if we didn't put in those parentheses, we would have definitely made some mistakes.2118

It is absolutely critical to put in parentheses when we are substituting.2121

Otherwise, mistakes will just start popping up everywhere.2125

All right, I hope all that made sense; we will see you at later--goodbye!2128