introduction to algebra module1
You have to understand that, just as with any language, like English or Afrikaans or Spanish, there is also a need for a “Mathematics Language” that we can use when doing maths. We will start by laying down some of the rules of this language. Let’s firstly look at some practical examples: The area of a rectangle is: Width Length Area = Length X Width We could calculate the area of any rectangle, using this formula. Do you agree? Do it then for the following rectangles: 10
8
5
12
50
5 What is the total area of the following two rectangles? 4
8 5 10
We will encounter a “language” problem somewhere soon, though! All the words that we used so far: “area” “length” “width” are in English. For a person in Timbuktu it may mean just as much as the following would mean to us: “” “” “” Therefore we must use symbols that would be understandable for everyone, and we then say that the Area of a rectangle is: A=l X b Where everyone in the world will know that “A” is for “Area”, l is for the long side and, and b is for the short side. If we look again at the initial two rectangles (in light green) and we also state that a= 5 b=4 we could say that the first rectangle’s length was a and its width was b b
A=aXb
a The second rectangle’s length was 10, and therefore we can say that it was 2 X 5 or 2 X a The same goes for the width, which was 8, or 2 X b.
The second rectangle will then look like this:
2b A = 2Xa X 2Xb 2a We do not have to use the “x” sign in Algebra, and we can then write the formula like this: A = 2a2b Now we need to pin down a few rules: 1) We have A = 2a2b, which means it is four terms that are multiplied. We know that 2 times by 2 is equal to 4, so we can write this as A = 4ab 2)
If we want to add the areas of the two green rectangles, we can write it as follows: A TOTAAL = ab + 4ab These two “things” are the same, and will give us 5ab. Do you agree? Then A = 5ab
3)
We can now take the dimensions of any rectangle, and substitute it into l and b, and the area will always be calculated with the formula A = l X b, and we can also do calculations with these values, such as adding the areas of different rectangles together. The values of l and b can change according to the rectangle that we are working with
We then call these letters variables, for the simple reason that their values are not fixed, but could “vary” according to where and how it is used. So, remember, there are rules that you need to adhere to, and if you know and understand these rules, the language of Mathematics will not be difficult. When we look at something like 4x + 3, we call the (as mentioned before) x a variable, because its value could be whatever we want to substitute into it, but the 3 is always 3 – never 4 or 20 or 100, and therefore the 3 is known as a constant. The 4 in front of the x is actually also a constant, but since it is being multiplied with the x (remember that there is actually a multiplication between these two terms) we have a special name for it: coefficient. We always write the coefficient (constant) in front of the variable, i.e. not x5 but 5x. This is different from quotient! If the coefficient is “1”, it does not have to be written down i.e. 1x = x When you really start working with algebraic expressions – once you understand the need for, and rules of, variables – there will also be some more terminologies that you must know, such as the following: 1. The sum of x and 3 is written as x+3 2.
The product of 5 and x is written as 5 X x = 5x
3.
Term: Only + or – separates terms (not a + / - in a set of brackets)
Examples 1) 14 x 2 12 y 6 Number of terms: Constant term: Variables: Coefficient of x 2 : Coefficient of y : 2)
3)
4)
y2 4 x y2 x 6 4 Number of terms: Constant term: Variables: Coefficient of x 2 : Coefficient of y 2 :
________ ________ ________ ________ ________
2
________ ________ ________ ________ ________
How many terms are there in the following algebraic expressions? a) 5x + 4y – 3z b) 2 x a x b x c c)
a b 2a 5 3
d)
3b 2 2a 5
e)
5a 3b
f)
(x – 1)(x – 2)
1 2a
Write down the coefficient of the variable x in each of the following: 2 a) 3 x 3 b) 7x – 5y + 1 c)
2xxx5
8
5
12
50
5 What is the total area of the following two rectangles? 4
8 5 10
We will encounter a “language” problem somewhere soon, though! All the words that we used so far: “area” “length” “width” are in English. For a person in Timbuktu it may mean just as much as the following would mean to us: “” “” “” Therefore we must use symbols that would be understandable for everyone, and we then say that the Area of a rectangle is: A=l X b Where everyone in the world will know that “A” is for “Area”, l is for the long side and, and b is for the short side. If we look again at the initial two rectangles (in light green) and we also state that a= 5 b=4 we could say that the first rectangle’s length was a and its width was b b
A=aXb
a The second rectangle’s length was 10, and therefore we can say that it was 2 X 5 or 2 X a The same goes for the width, which was 8, or 2 X b.
The second rectangle will then look like this:
2b A = 2Xa X 2Xb 2a We do not have to use the “x” sign in Algebra, and we can then write the formula like this: A = 2a2b Now we need to pin down a few rules: 1) We have A = 2a2b, which means it is four terms that are multiplied. We know that 2 times by 2 is equal to 4, so we can write this as A = 4ab 2)
If we want to add the areas of the two green rectangles, we can write it as follows: A TOTAAL = ab + 4ab These two “things” are the same, and will give us 5ab. Do you agree? Then A = 5ab
3)
We can now take the dimensions of any rectangle, and substitute it into l and b, and the area will always be calculated with the formula A = l X b, and we can also do calculations with these values, such as adding the areas of different rectangles together. The values of l and b can change according to the rectangle that we are working with
We then call these letters variables, for the simple reason that their values are not fixed, but could “vary” according to where and how it is used. So, remember, there are rules that you need to adhere to, and if you know and understand these rules, the language of Mathematics will not be difficult. When we look at something like 4x + 3, we call the (as mentioned before) x a variable, because its value could be whatever we want to substitute into it, but the 3 is always 3 – never 4 or 20 or 100, and therefore the 3 is known as a constant. The 4 in front of the x is actually also a constant, but since it is being multiplied with the x (remember that there is actually a multiplication between these two terms) we have a special name for it: coefficient. We always write the coefficient (constant) in front of the variable, i.e. not x5 but 5x. This is different from quotient! If the coefficient is “1”, it does not have to be written down i.e. 1x = x When you really start working with algebraic expressions – once you understand the need for, and rules of, variables – there will also be some more terminologies that you must know, such as the following: 1. The sum of x and 3 is written as x+3 2.
The product of 5 and x is written as 5 X x = 5x
3.
Term: Only + or – separates terms (not a + / - in a set of brackets)
Examples 1) 14 x 2 12 y 6 Number of terms: Constant term: Variables: Coefficient of x 2 : Coefficient of y : 2)
3)
4)
y2 4 x y2 x 6 4 Number of terms: Constant term: Variables: Coefficient of x 2 : Coefficient of y 2 :
________ ________ ________ ________ ________
2
________ ________ ________ ________ ________
How many terms are there in the following algebraic expressions? a) 5x + 4y – 3z b) 2 x a x b x c c)
a b 2a 5 3
d)
3b 2 2a 5
e)
5a 3b
f)
(x – 1)(x – 2)
1 2a
Write down the coefficient of the variable x in each of the following: 2 a) 3 x 3 b) 7x – 5y + 1 c)
2xxx5
