00:02 Hello.

00:14 My name is Dorcas Awaye,

00:16 and today I'll be taking you mathematics.

00:18 Today we'll be discussing the implication

00:20 of algebraic expressions.

00:22 At the end of today's teaching,

00:25 you should be able to identify and collect

00:26 like terms in a given expression.

00:29 You should be able to identify the coefficient

00:31 of a given algebraic term.

00:33 Identify the positive and negative coefficients

00:35 of a given algebraic term.

00:38 Perform basic arithmetic operations

00:40 on expressions of similar terms.

00:43 Insert and remove brackets and simplify expressions.

00:48 Solve quantitative attitude problems

00:49 on the use of brackets.

00:51 Algorithmic expressions and terms.

00:54 An algebraic expression is a mathematical statement

00:57 that contains a combination of numbers.

01:01 Symbols variable such as X or Y, A and B,

01:06 and mathematical operators such as the addition sign,

01:09 subtraction, multiplication, or division.

01:13 It doesn't have equal sign.

01:15 Just like we did in previous classes,

01:18 we talked about open sentences where we have symbols

01:22 in addition to numbers where we have to look for the unknown

01:26 and then we moved on to talking about using alphabets

01:31 to represent symbols which are either triangles or squares.

01:36 Here we deal with alphabets majorly,

01:40 such as X or Y to represent the unknown values.

01:44 Here are some examples of algebraic expressions.

01:49 We have two X plus 10 Y plus one.

01:53 You can note that it does not have an equal sign.

01:56 When it has an equal sign, that makes it an equation.

01:59 But this is just an expression.

02:02 It has a combination of numbers attached to alphabets.

02:06 So here, the exact value of two X is unknown

02:11 because we don't know what value X is taking at the time.

02:15 Remember, we talked about variables in the last class

02:18 and then X is an example of a variable.

02:21 Y is an example of a variable.

02:23 Another example is minus 3m minus 7n plus 2.

02:26 This is also an algebraic expression.

02:28 I'll move on to terms.

02:32 Algebraic expressions are made up of terms

02:34 that are separated by an addition or subtraction sign.

02:38 The algebraic expression, 2x plus 1, asks 2 times.

02:42 From this expression, looking at 2x plus 1, how did I

02:50 come about 2 times?

02:51 Remember, I told you what terms mean.

02:53 They are made up of numbers which are separated by either

02:57 addition or the subtraction sign.

02:59 Looking at this, you may think this has just one term

03:02 because we have minus 1.

03:04 But there is an invincible plus 2 here that makes it a term.

03:08 So this is one term.

03:09 This is another term.

03:10 So we have two terms here.

03:14 So for this algebraic expression, we have two terms.

03:19 We can have more than two terms.

03:21 For example, you have 2x plus 3 plus 5y.

03:27 You don't have to be in an algebraic expression

03:29 for it to become terms.

03:31 So long, it is separated by either an addition or subtraction

03:36 or pre-talk, it becomes a term.

03:37 So for example, this is term 1.

03:39 This is term 2.

03:40 This is term 3.

03:42 A term can be a single number, positive or negative.

03:46 You can have minus 3 plus 2.

03:50 Here, we have two terms.

03:52 This is a term.

03:53 This is another term.

03:55 Plus 2 is another term.

03:59 A term can be a single number, positive or negative,

04:01 or a variable, one or more numbers and variables

04:06 that have been multiplied together,

04:09 terms by terms, and are separated by addition or subtraction

04:14 sign.

04:16 Here are some more examples of single terms.

04:18 This is a single term, 3xyz, 5c minus 7x squared,

04:24 minus 7x squared.

04:25 These are single terms.

04:30 Or algebraic expressions with two terms

04:32 include 2y plus 5, minus 5y, minus 10b.

04:37 Walking with algebraic expressions, when

04:44 you're lent to hard or subtract objects,

04:47 you lent that, for example, one apple plus one apple plus one

04:51 apple plus one apple can be written as four apples.

04:54 That is one apple, one apple plus another one.

05:03 Let me represent it as--

05:05 let me just keep writing like this--

05:06 plus another one plus another one apple plus another one

05:12 can also be written as four apples.

05:16 Just the way we did during counting with symbols,

05:20 we can have boxes plus boxes, a box plus another box.

05:25 This would give you three boxes.

05:29 So it is when you are adding and subtracting algebraic expressions.

05:34 You can as well be, for example, for four apples.

05:37 Instead of just one plus one plus one,

05:39 you can still say four times one apple.

05:44 It is the same thing.

05:47 It gives you the same four apples.

05:51 So that means, apple in four places,

05:54 that's four times over apple, gives you four apples.

05:57 It is the same thing as adding apples four times.

06:00 The same applies to adding numbers.

06:07 9 plus 9 plus 9 plus another 9 can also

06:11 be written as four times 9, which gives you the same result.

06:14 9 in four places gives you 36.

06:18 9 in four or four in nine places gives you 36.

06:22 The same principles apply in algebra.

06:24 But remember that in algebra, we use a letter of variable

06:27 to represent a number.

06:29 For example, a plus a plus a plus a is the same thing as four

06:34 multiplied by a.

06:37 In algebra, we normally do not write the multiplication sign

06:40 between a number and a variable.

06:42 For example, we write four times a as four a.

06:46 If you have two a, it is the same thing as two times a.

06:54 But in algebraic expressions in mathematics,

06:57 we don't write two times a.

06:59 Instead, we write two a.

07:01 It is easier that way.

07:03 So when you see five b, you see two c.

07:07 You know that it is five multiplied by b.

07:10 Two multiplied by c. That is what it means.

07:13 So the following expressions using a simple annotation.

07:17 Y plus y plus y plus y.

07:19 Instead of writing all of that, you just have five y.

07:23 The same thing with 8 times b gives you 8 b.

07:25 Two a in three places, 6a.

07:28 Simple.

07:29 We move on to constants and coefficients.

07:33 The following algebraic expression contains two times minus four x plus five.

07:39 Remember, I told you what terms are earlier.

07:41 They are numbers or variables with signs.

07:47 Either addition sign or subtraction.

07:50 So looking at minus four x plus five, we have 10, 1 and 10, too.

08:01 But now we are talking about constants and coefficients.

08:04 What is a constant?

08:05 What is a coefficient?

08:06 In this expression, five is a constant.

08:09 How is it a constant?

08:10 Because it does not change its value.

08:12 It continues to be five today, tomorrow, next tomorrow.

08:16 It doesn't change its value.

08:18 It does not have a variable attached to it.

08:20 So it cannot vary.

08:22 That is why it is called a constant.

08:25 For four x, this is not a constant term.

08:31 Because it has a variable attached to it.

08:34 So you cannot regard this.

08:36 You can't regard this as a constant.

08:39 It is a variable term.

08:41 So we have constant and variable.

08:47 Now, to say further on that, I'm

08:55 going to talk about coefficients.

08:57 Coefficient is the number or factor that

09:00 has been multiplied by the variable.

09:03 In this expression, minus four x, the coefficient is minus four.

09:12 In this expression, the coefficient is minus four, just by the definition.

09:16 Coefficient is the number or factor being multiplied by the variable.

09:21 A coefficient can have a negative value just as it is here.

09:25 If it has a minus sign in front of it,

09:27 or a positive value, if it has a plus sign or no sign in front of it.

09:31 So where you see no sign, it means you have an invisible addition value,

09:35 addition operator.

09:37 It's a plus sign.

09:38 Without negative, it is positive.

09:41 So here we have minus four x.

09:43 And here, the coefficient is minus four, while this is a constant.

09:53 Another example, if you have three x plus two, this is your constant.

10:02 And this is three of three x is the coefficient.

10:06 So the coefficients of x in this expression is three, or rather plus three.

10:12 You can say three or plus three.

10:17 Looking at this image, three x plus five y minus nine.

10:21 So we can easily note which ones are the coefficients.

10:25 For the variable x, the coefficient is three.

10:28 For variable y, the coefficient here is plus five.

10:31 And then we have constant of minus nine.

10:35 x and y are the variables and the terms.

10:37 We have three terms here.

10:39 We have three x, five y, and then nine.

10:45 We have worked examples.

10:47 For the algebraic expressions below, write down the number of terms.

10:51 The constant term, the coefficient of x, the coefficient of y, and

10:56 the coefficient of k.

10:57 So let's just get to it quickly.

10:59 Two x, the first one, two x plus y, minus four k, minus six.

11:12 How many terms do we have here?

11:13 We have one, two, three, four.

11:18 Why do we have y as part of the terms?

11:21 Because, of course, we have an invisible one here.

11:23 So it doesn't stand on its own.

11:25 Remember, I told you a term can just be a variable that has

11:28 an additional subtraction, but it doesn't have to be a number

11:33 that you are used to maybe two y or y.

11:36 So since it is y, it's also known as one y.

11:39 Do you understand?

11:40 So for here, we have many terms.

11:42 We have one, two, three and four terms.

11:48 The next question is, what are the constant terms here?

11:51 The constant term is six because it doesn't have any variable

11:56 attached to it.

11:57 So this is constant.

11:59 The constant is six.

12:00 The coefficient of x, the coefficient of x here is two.

12:08 Coefficient of x is two.

12:14 Then the coefficient of y, coefficient of y, of y is one.

12:33 The coefficient of k of k is minus four.

12:46 The same thing applies to the second example.

12:48 The second question, 10 minus x plus two y.

12:52 So looking at this, the coefficient of y is plus two.

12:58 The coefficient of x is one.

13:01 There's an invisible one here.

13:03 You don't just have x, x on its own is the same thing as one x.

13:07 So if anybody acts you, what is the coefficient of x?

13:11 You say one.

13:13 It is invisible, but we always know that it is there.

13:16 And then the constant here is 10 for the second example.

13:22 Add in unsubtracting light terms.

13:25 We have seen that we can add p plus p plus p to get 3p.

13:30 These terms are called light terms.

13:33 Why?

13:34 Because they contain the same variables, which is p.

13:43 We can also add 3x plus 4x.

13:45 Because both terms have the variable x.

13:49 We can combine the terms using the operation that is given,

13:52 which is in this case, the addition.

13:54 The correct answer here is 7x, 3x plus 4x is 7x.

13:59 Why?

13:59 Because they both have light terms.

14:01 They have light variables.

14:04 We can also subtract light terms.

14:07 We can simplify expressions like 10a minus 2a using subtraction.

14:11 The correct answer here is 8a.

14:13 So simplify the expression 2x plus y plus 3y.

14:21 So I'll just stop this now.

14:23 2x plus 3y plus 5x plus 2y.

14:34 How do you solve this?

14:36 The first thing you do is to rearrange.

14:40 In such a way, where some variables are together

14:43 and the others are on the other side, this way.

14:46 So solution to this, the first thing is rearrange.

14:54 So we have 2x.

14:55 Cut out the x's together.

14:56 2x plus 5x plus 3y plus 2y.

15:06 Now, 2x plus 5x can be the same thing as 2a plus 5a.

15:13 So 2x plus 5x is 7x.

15:15 Assuming they are apples.

15:19 2a plus 5a plus 7a.

15:23 3y plus 2y.

15:25 3a plus or 3 oranges plus 2 oranges.

15:28 We have 5 oranges.

15:29 So 3a plus 2 is 5.

15:30 Because they have the same terms, we can add them together.

15:33 So we have 5y.

15:36 Here, we leave the expression the way it is,

15:39 because it cannot be further simplified like this,

15:44 because there are no other terms.

15:45 We can add together or subtract from each other.

15:48 So this is where this stops.

15:49 This is the end of this solution.

15:54 Simplify the expression, 8x plus the tin y.

15:57 So I'm just going to do the next one quickly.

16:03 8x plus the tin y minus x minus 4y.

16:13 Just as I did earlier, you rearrange,

16:18 gathering those ones with x together, 8x minus x plus 13y minus 4y.

16:33 So 8x minus x.

16:35 8x is minus 1x.

16:37 Remember, there's an invisible one.

16:38 So you have 8 oranges minus 1 orange.

16:41 Just like that, 8x minus x is 7x plus 13y minus 4y is 9y.

16:51 So these cannot be further simplified,

16:53 because there are no other liking variables or terms,

17:00 as the case may be.

17:04 Walking with brackets in algebraic expressions.

17:07 We sometimes use brackets to group terms in algebraic expressions together.

17:11 For example, 2c is the same as 2 times c,

17:16 which is also the same as c plus c.

17:19 If we replace c with a plus b into bracket,

17:22 we have 2 a plus b equals 2 times a plus b.

17:27 Let me just show you briefly how this is expressed.

17:33 So 2 into bracket a plus b.

17:36 So this can be regarded as a term on its own,

17:39 which is the same thing as 2 times a plus b.

17:44 It is also the same thing as a plus b plus a plus b,

17:54 which is a plus b plus a plus b.

18:01 The same as, then you rearrange.

18:03 Like we did earlier, a plus a plus b plus b, which is 2a plus 2b.

18:11 How did I come about this?

18:12 This is 1a.

18:13 This is 1a.

18:14 This is 1b.

18:15 This is 1b.

18:16 That is how this came about.

18:17 So that's how we got 2a plus 2b.

18:22 Question 4m is the same as 4 times m.

18:24 You replace m with a plus b, then remove the brackets

18:27 and simplify the expression.

18:29 Here we use repeated addition.

18:31 So what we've been doing here is called the use of repeated

18:34 addition to work out the multiplication.

18:39 Use a distributive law to open brackets.

18:42 When we have an expression with brackets,

18:45 we know we need to multiply.

18:47 When we do this, we say that we expand the expression.

18:51 It is called expansion.

18:53 The simpler way, instead of repeated additions,

18:56 is to multiply the coefficients of the brackets

18:59 by each of the times that is inside the brackets.

19:03 When we do this, we are applying the distributive law.

19:06 This law states that 4 times 3 plus 2 is the same as 4 times 3

19:12 plus 4 times 2.

19:14 In other words, 4 groups of 3 plus 2

19:18 is the same as 4 groups of 3 plus 4 groups of 2.

19:23 For example, we expand this expression.

19:27 3 into brackets 2a plus b.

19:36 Now with expansion and the distributive law, this is--

19:40 you are going to use this to multiply this, then you add,

19:48 and then use this same to multiply this.

19:51 So we have 3 times 2a can just bracket first plus 3

20:00 multiplied by b.

20:03 So we have 3 times 2a is 6a, 3 times 2.

20:10 This expression is the same as 3 times 2 times a.

20:17 Remember 2a is the same as 2a.

20:19 So in our algebraic expression, you

20:21 can multiply all these together.

20:23 3 times 2 times a.

20:25 Note you cannot add 3 plus 2 plus a.

20:28 You are not like 10s.

20:30 So 3 times 2 times a is 6a plus 3 times b is 3b.

20:39 In our algebra, 2 times a is 2a.

20:49 So the first term is 6a.

20:52 Like it says here.

20:53 Multiply 3 by the second term, like I explained earlier,

20:56 and you get 3b.

20:57 At the end of the day, you get 6a plus 3b.

21:00 More examples.

21:02 You can solve this on your own.

21:03 I'll just solve just one of it.

21:04 Expand the following expression, 2a plus.

21:07 So we have 2 into bracket a plus 5b minus 3c.

21:17 Like I said earlier, you use 2 to multiply f

21:22 first of all plus 2 times 5b.

21:30 The sign just follows minus 2 times 3c.

21:38 So it's just use this to multiply this at first.

21:42 Put your addition.

21:43 The same this multiplied this, include your operative sign,

21:48 and then you multiplied this as well.

21:51 So the answer to this is 2 times a is 2a plus 2 times 5b

21:56 is the same thing as 2 times 5 times b.

22:09 Minus 2 times 3 times c.

22:16 So 2a plus 2 times 5 is 10.

22:23 10 times b is 10b minus 2 times 3 is 6.

22:29 6 times c is 6c, 6c.

22:33 So we have 2a plus 10b minus 6c.

22:39 So this is the answer to this.

22:41 I believe you should be able to solve the rest.

22:43 I'm just going to go through one more example

22:47 that you may find a bit difficult, which is a third example.

22:51 Minus 2 into bracket 3m plus 5.

22:58 Looking at this, you have a negative multiplier here.

23:03 So just the same way you've been doing it minus 2 into multiply

23:08 this and then to multiply this as well.

23:10 So you have minus 2 into bracket 3m plus we have minus 2.

23:18 So you bracket it out.

23:20 5.

23:25 So minus 2 times 3m is minus 6m.

23:30 Now you should remember the rules of multiplication,

23:34 the rules you use in multiplying addition and subtraction.

23:39 So minus times minus is plus plus times minus is minus

23:46 is the same thing as minus times plus,

23:49 which is the same thing as minus.

23:51 So here plus, remember that it is multiplying

23:54 what is in the bracket plus times minus is minus 2.

23:59 So minus 2, let me just go from here, minus 2 times plus 5

24:02 should give you minus 10.

24:04 So 2 times 5 is 10 here.

24:08 We have plus sign here, that's minus 10.

24:11 Remember plus times minus is minus and you have 6m, minus 6m,

24:15 plus times minus minus 10.

24:18 So this cannot be solved any further,

24:20 because we do not have a variable where you can subtract m together.

24:28 So it's 6m, minus 10, minus 10 is where we stop

24:32 for this order of operations.

24:36 In the previous classes, we discussed order of operations.

24:40 And we have the acronym BUDMAS.

24:42 Because the first thing you do when

24:44 you see a large expression filled with brackets, multiplication,

24:49 addition, and subtraction, the first thing you

24:51 try to solve is the brackets.

24:52 You go then with the off, division, multiplication,

24:56 addition, and then subtraction.

24:58 Work the example.

25:00 Simplify the expression 2 into bracket 3x plus 5x plus 1,

25:04 plus 4 times 3.

25:06 The first step is to simplify the terms inside the bracket

25:09 first.

25:09 Just as it is, the bracket goes first.

25:13 Step 1, simplify the terms inside the bracket first.

25:17 That is minus 2 into bracket 8x plus 1, plus 4 times 3x.

25:22 That is, you have multiplied.

25:23 You've already added 3x plus 5x, which gave you 8x.

25:27 That is 8x plus 1.

25:29 Step 2, because there is no off, which

25:31 is the next one, or division.

25:34 In the expression, you do multiplication next.

25:37 So you use the distributive law to multiply the brackets

25:40 by its coefficient.

25:42 Remember that 4 times 3x means 4 times 3 times x.

25:47 16x plus 2 plus 12x.

25:50 Then you collect the light terms.

25:52 16x plus 2x plus 2 is 28x plus 2.

25:58 So that is how this ends.

26:01 We can add the other expression.

26:03 2y plus 6y divided by 2 times 2 in brackets.

26:07 And then times 10 minus 5 times times 2y plus another 2y.

26:12 So to simplify this, you just go along

26:14 with the steps as same here.

26:18 Your assignment, 5y minus 4y minus 3y,

26:21 using the distributive law, using the laws of multiplication,

26:25 I believe you should be able to solve this easily.

26:29 So if you come to the end of today's lesson,

26:31 thank you very much for watching.

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