### The Vocabulary of Linear Equations

- A variable is a letter that is used to represent any unknown quantity. Usually we use the letter “x.”
- A term is a number, variable, or product or quotient of numbers and variables raised to powers. Terms can be identified because they are connected to other terms using addition or subtraction.
- Terms with exactly the same variables that have the same exponents are called like terms.
- An equation is a statement that two algebraic expression are equal. A linear equation is a special type of equation that can be written in the form Ax + B = C where A, B, and C are real numbers with A not being zero.
- A number is said to be a solution if it can be substituted for the variable, and it creates a true statement.

### The Vocabulary of Linear Equations

Six times a number decreased by the square of that number is three more than five times the number.

- x represents the unknown number

**6**

**x**−

**x**

^{2}

**=**

**5**

**x**

**+**

**3**

Eight times the sum of a number and the cube of another number is ten less than the difference of the second number and four times the first number.

- x = 1st number

y = 2nd number

**8**

**(**

**x**

**+**

**y**

^{3}

**)**

**=**

**(**

**y**−

**4**

**x**

**)**−

**10**

- x represents the unknown number

**12**

**x**

**+**

**x**

^{3}

**=**

**8**

**x**

**+**

**20**

Sixteen times the difference of a number and half of another number is three less than the sum of the square of the second number and seven times the first number.

- x = 1st number

y = 2nd number

**16**

**(**

**x**− [(

**y**)/(

**2**)]

**)**

**=**

**(**

**y**

^{2}

**+**

**7**

**x**

**)**−

**3**

John's brother is ten years younger than he is. The product of their ages is seventeen less than the square of the difference of their ages.

- j = John's age

j - 10 = John's sister's age - j(j − 10) = [j − (j − 10)]
^{2}− 17 - j(j − 10) = (j − j + 10)
^{2}− 17 - j(j − 10) = 10
^{2}− 17 - j(j − 10) = 100 − 17

**j**

**(**

**j**−

**10**

**)**

**=**

**83**

Jason's friend is six years older than he is. The difference of their age is three times Jason's age.

- j = Jason's age

j + 6 = Jason's friend's age

**j**−

**(**

**j**

**+**

**6**

**)**

**=**

**3**

**(**

**j**

**+**

**6**

**)**

- x represents the unknown number

**x**

^{2}−

**10**

**x**

**=**

**5**

**x**−

**2**

Half of a number is seven more than four times the square of that number.

- x represents the unknown number

**x**)/(

**2**)]

**=**

**4**

**x**

^{2}

**+**

**7**

Nine times the difference of a number and the cube of another number is seven more than the difference of two times the square of the second number and three times the first number.

- x = 1st number

y = 2nd number

**9**

**(**

**x**−

**y**

^{3}

**)**

**=**

**(**

**2**

**y**

^{2}−

**3**

**x**

**)**

**+**

**7**

Sarah's sister is twice her age. The sum of their ages is three less than the square of the difference of their ages.

- s = Sarah's age

2s = Sarah's sister's age

**s**

**+**

**2**

**s**

**=**

**(**

**s**−

**2**

**s**

**)**

^{2}−

**3**

*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.

Answer

### The Vocabulary 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 0:09
- The Vocabulary of Linear Equations 0:44
- Variables
- Terms
- Coefficients
- Like Terms
- Examples of Like Terms
- Expressions
- Equations
- Linear Equations
- Solutions
- Example 1 6:16
- Example 2 7:16
- Example 3 8:45
- Example 4 10:20

### Algebra 1 Online Course

### Transcription: 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 3x ^{2} and 4x^{2}, 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 y ^{2} 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 2x ^{2}.*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 + (y ^{2} + 6y^{2}), 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 y ^{2}.*0675

*Let us put those together, 1y - 3y would be -2y, I have a 4 hanging by itself, -4.*0683

*y ^{2} + 6y^{2} would be a 7y^{2}, 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 + 14y ^{2}.*0708

*My final answer would be -4y - 8 + 14y ^{2}.*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 y ^{2}, definitely not like terms.*0735

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

1 answer

Last reply by: Professor Eric Smith

Mon Jun 16, 2014 3:00 PM

Post by Timothy Davis on June 10, 2014

I am thoroughly enjoying your lectures! I have always struggled with calculus and differential equations, and after watching your videos up this point, I realize it is because my algebra 1 skills are so weak!. Thanks!

1 answer

Last reply by: Professor Eric Smith

Tue Oct 15, 2013 10:15 AM

Post by Emma Wright on October 14, 2013

So, When You had 20w-9+3 why did it end up as 20w-6?

Also, why when you subtract y-3y why is it 2, and when you add y2+6y2 it ends up as 7y2 ?

3 answers

Last reply by: Professor Eric Smith

Tue Oct 15, 2013 10:16 AM

Post by Ezuma Ngwu on August 29, 2013

How do you create and solve equations with tiles?