Algebra 1 The Vocabulary of Linear Equations

Welcome back to www.educator.com.0000

In this lesson we are going to start looking more at linear equation starting off with the vocabulary of linear equations.0003

There will be lots of new terms in here, it will definitely take some time to look at them all and what they mean and play around with them a little bit.0010

Some of the terms that will definitely get more familiar with are variable, term.0018

We will look at coefficients and we will definitely see how we can combine like terms.0026

We will be able to tell the difference between equations and expressions and get into a linear equation.0032

What we want to solve later on and solutions.0038

When looking at an equation, we often see lots of letters in there, those are our variables.0046

What they do is they represent our unknowns.0054

One favorite thing to use in a lot of equations is (x), but potentially we could use any letter.0058

You could use a, b, it does not really matter but most the time our unknown is (x).0064

A term is a little bit more than just that variable.0070

It is a number of variables or sometimes the product or quotient of those things put together.0074

To make it a little bit more clear, I have different examples of what I mean by a term.0079

All of these things down here are types of terms.0085

The first one is the product of an actual number and a variable.0088

Down here with the (k) it is simply just a variable all by itself.0093

A coefficient of a term is a number associated with that term.0102

If I'm looking at a term say 2m, the coefficient is the number right out front that is associated with that term.0109

I'm looking at another one like 5mq, but again the 5 would be the coefficient of that term.0121

Terms with the exactly the same variables that have the same exponents those are known as like terms.0133

There are two conditions in there you want to be familiar with.0140

It must have exactly the same variables and it must also have exactly the same exponents.0143

Let us say I have both of those and you can not consider them like terms.0150

Let us take a look at some real quick.0156

I'm looking at 5x and I'm looking at 5y, these are not like terms.0158

Why, you ask? They do not have the same variable.0169

One has (x) and the other one has (y).0172

How about 3x² and 4x², these are like terms.0176

These ones are definitely good because notice they have exactly the same variable and they have the same exponent.0188

They have both of those conditions.0203

This one is little bit trickier so be careful, they both have an (x), that looks good.0218

They both have a (y), that seems good but they have different exponents.0224

This one has the y² and this one has nothing on its y.0230

I would say that these are not like terms.0238

An expression is the statement written using a combination of these numbers, operations, and variables.0244

This is when we start stringing things together so I might have a term 2x and then I decide use may be addition and put together a 4xy.0251

That entire thing would be my expression.0262

In the equation, we take a statement that two algebraic expressions are actually equal.0265

I can even borrow my previous expression to make an equation.0271

I simply have to set it equal to another expression, maybe 2x².0276

What a lot of students like to recognize in these two cases is that in an equation there better be an equal sign somewhere in there.0285

With your expression there is not an equal sign because it is just a whole bunch of string of terms and coefficients, numbers, operations.0292

Since we are interested in linear equations and eventually getting solved for those, what exactly is a linear equation?0306

It is any equation that can be written in the form, ax + b = c.0314

There is some conditions on those a, b, and c.0321

Here we want a, b, and c to be real numbers, we would not have to deal with any of those imaginary guys.0324

We want to make sure that (a) is not 0.0330

The reason why we are throwing that condition in there is we do not want to get rid of our variable.0335

If (a) was 0, you would have 0 × x and then we would have a rare variable whatsoever.0340

It is an equation, not necessarily look like that but it almost can be written in that form, it is a linear equation.0347

A number is a solution of that equation if after substituting it in for the value the statement is true.0354

That means if you actually take out your variable and replace it with a number that is the solution.0361

Then it is going to balance out, it is going to be true with that number in there.0367

This first part we just want to identify the different parts of the equation so we can better feel of what we are looking at.0379

First of all I know that this is an equation because notice how we have an equal sign right there.0386

I have the expression 7x + 8 that is one expression one side and expression 15 on the other.0394

What else do I have here? I have my 7x and I have the 8 both of these are terms.0400

If I pick apart that one term on the left, I can say that the 7 is a coefficient and that the x is my variable.0410

There are many different parts of the equation and you want be able to keep track of it.0428

And probably terms are one of the most important for now.0432

Let us look at this one, let us see 30x = (4 × X) - 3 + (3 × 3) + (x + 2).0438

Again identify the parts of the equation, let us see.0444

It has the equal sign, I know it is an equation, it is important to recognize.0449

Let us see, over on this side I have a term 30x, I have this term, I have this term.0454

I have a bunch of different terms.0461

Terms are always combined using addition or subtraction.0465

That is how we can usually recognize them.0468

In my terms I have some coefficients but you know it might be easier to use my distributive properties to see even more of those coefficients.0471

Let us use that distributive property, let us take the 4 multiplied by the x and 3 and do the same thing with 3.0481

4x -12 + 3x + 6, not bad.0489

Looking at that I have even more terms, I got my 30x, I get 4x, 12, 3x, 6 and lots of different terms now.0497

Into those terms I can identify what its coefficient is and I can identify the variable, the x.0508

This one says simplify it by combining like terms.0528

Remember our like terms are terms that have exactly the same variable and they have exactly the same exponent.0532

We have to be careful on which things we can actually combine here.0539

Let us see I have 12w and 10w those are like terms, they both have a w to the first.0543

Over here is 8- 2w, all three of those are like terms, we only write them next to each other.0550

I know that I need to combine them.0558

The 9 and the 3 I would also consider those like terms because both of them do not have a variable associated with them.0567

I will combine those together.0574

Let us take care of everything with the (w), 12w + 10 would be 22w.0576

That would give me a 20w when combining those, now we will take care of 3 and 9, -3 + 9 = - 6.0591

It is important to recognize that you should not go any further than here because the 20w and 6, those are not like terms.0609

We are not going to put those together.0616

On to example 4, this one we want to simplify by combining like terms.0622

I have lots of grouping symbols in here so it is hard to pick out what my like terms are.0630

I think we can do it though but we might have to borrow our distributive property first.0637

I'm going to take this negative sign and distribute it inside my parentheses here.0641

Now that would give me (2y - 3y) - 4 + (y² + 6y²), not bad.0648

I can see there is a few things I can combine, let us see.0665

Specifically I can put these y's together since both of them are single (y).0670

I can combine these ones together over here because both of them are y².0675

Let us put those together, 1y - 3y would be -2y, I have a 4 hanging by itself, -4.0683

y² + 6y² would be a 7y², that looks good.0695

Remember, I have not distributed my 2, it is not going to help me combine anymore like terms.0703

It will definitely help me see my final results, -4y - 8 + 14y².0708

My final answer would be -4y - 8 + 14y².0722

I would not combine those anymore together because none of those are like terms.0726

I have a single (y), I have an 8 that does not have any variables whatsoever.0731

I have that y², definitely not like terms.0735

Alright, thanks for watching www.educator.com.0739