Algebra-cadabra - the magic of maths
Part 2 of David's Algebra for beginners
In the previous article, we discussed the use of numbers and letters, and had a look at some of the terminology that is used in algebra. We also looked at some of the mathematical operators that are used in mathematics. As you can imagine, the four main operations (adding, subtracting, multiplying, and dividing) are used very frequently.
In this article, we will start by looking at how multiplication is used in algebra, and the way it is written down. We'll then look at addition and subtraction. Division is a slightly more complicated subject that we'll leave until a later article.
As before, there are exercises on the various topics for you to try at the end.
First of all, two more bits of terminology
It is often sensible to simplify an expression as much as possible. This makes it more manageable, and usually easier to evaluate if we are given values for the letters. For example, the expression 2x + 4y + 16z - y + 2z - x - 17z - 2y is pretty cumbersome. By adding and subtracting the 'like terms' we can simplify this expression to x + y + z, which is much easier to handle. I'll explain later how this is done.
If we know the value that each letter holds, we can substitute those values into the expression, and evaluate it. For example, the expression above has been simplified to x + y + z. If we know that x = 10, y = 25, and z = 1, we can substitute those values and work out the answer, which is 36.
With algebra there can be a bit of a problem with multiplication, since the multiplication sign, x, can be confused with the letter x. For this reason the asterisk can be used instead.
In algebra it is normal to omit the multiplication sign altogether. However, this convention is only used where letters are used, or where letters and numbers are combined. Where just numbers are used, the multiplication sign is included.
Here are some examples
If we have just numbers, we write
3 * 5
10 * 4
2 * 4
and where we have more than two numbers, we would write
2 * 3 * 5 * 10
Where we have just letters...
x * y is written xy
a * b is written ab
P * Q is written PQ
If there there is more than one letter...
a * b * c is written abc
Normally, the letters are kept in alphabetical order.
Where we have numbers and letters...
5 * x is written 5x
8 * z is written 8z
25 * P is written 25P
The number is always written first, so that
a * 20 is written 20a
Where there is more than one letter and one number...
x * y * 9 * z is written 9xyz
Where there are letters and more than one number, we would normally multiply the numbers together, and put the result with the letters, or, if it is more appropriate, we can leave it unresolved. For example,
5 * 3 * x could be written as 15x, or 5 * 3x., or even 3 * 5x
The order of multiplication
With multiplication, it doesn't matter which order the letters and numbers are multiplied in.
For example, 3 * 5 * 2 is the same as 2 * 5 * 3, or 3 * 2 * 5, or 2 * 3 * 5. Whichever way you do it, the answer is always 30.
In the same way, 5 * x * y is the same as x * 5 * y, or y * x * 5, or 5 * y * x. Whichever way you do it, the answer is always 5xy
The coefficient of a term is the numerical value attached to it. Every term involving letters has a coefficient. The coefficient tells us 'how many' of that letter there are. Imagine a box of chocolate assortments. Let's say our box contains 15 orange cremes, 6 fudges, and 8 caramels. The coefficient of the orange cremes is 15. The coefficient of the fudges is 6. What is the coefficient of the caramels?
This can also be seen as a multiplication. You have 15 'lots' of orange creme. In others words, 5 * orange creme. Your chocolate box comprises 15 * orange creme, 6 * fudge, and 8 * caramel.
It works the same way with terms involving letters in algebra.
The term 5x means 5 lots of x, or 5 * x
The term 2a means 2 lots of a, or 2 * a
Even a letter by itself has a coefficient. x really means 1x, or 1 * x. In algebra, a coefficient of one is usually omitted, but it is always implied.
Addition and subtraction
Addition and subtraction are extremely common in all branches of mathematics, and algebra is no exception. Algebra, however, does have a peculiarity in that, because it makes use of letters and symbols which represent specific values, care has to be taken when adding and subtracting them.
In ordinary arithmetic, we can add and subtract any numbers, without problem. For example, we can add 3 and 3, and the answer is 6, every time. In algebra, we can only add or subtract similar terms. We can add x and x to make 2x. (Remember x is actually 1x). We can add y and y, to make 2y. But we cannot add x and y because they are different. In other words, x + y cannot be resolved. The expression must remain as x + y until we know the values that x and y represent. Then we can add them together to get a numerical answer
Consider the expression a + b + c.
If we don't know the values that the letters represent, the expression can't be simplified further, and it remains a + b + c.
However, if we are given that a = 10, b = 5, and c = 1, we can now evaluate the expression.
a + b + c = 16.
If we are given that a = 25, and b = 6, but we are given no value for c, then we can add the 25 and the 6 to give 31. So in this case, a + b + c can be simplified to give 31 + c, but that's all we can do.
The same difficulties arise with subtraction. Take the expression p - q - r. If we are given that p = 25, and r = 5, we can rewrite the expression as 25 - q - 5. By subtracting 5 from 25, this can be simplified to 20 - q, but no further.
One thing we can do to simplify complicated expressions involving addition and subtraction is to 'collect like terms'. We have already seen that like terms in an expression are those which have the same combination of letters, but not necessarily the same coefficients. Examples of like terms are 3p and 4p, 15x and 2x, 5abc and 7abc, etc. If we have an expression where there are a number of like terms, these can be added together or subtracted from each other, because they represent the same values.
For example, let's take an expression like
16a + 5b - 10a + 2a - b - 7a - 3b
The first thing we must notice is that each term has an operator symbol in front of it. This symbol always refers to the term following it, and it tells us whether that term is to be added or subtracted. In our expression, 5b is going to be added, while 10a is going to be subtracted. The first term in an expression normally doesn't have an operator, but unless we are told otherwise, we always assume it is +.
The expression can be simplified in the following way:
* collect the 'a' terms, and add or subtract them. This gives us 16a - 10a + 2a - 7a. This can be simplified to give a (actually, this is 1a, but we ignore the 1)
* collect the 'b' terms, and add or subtract them. This gives us 5b - b - 3b, which simplifies to b
* Combine the two results to give the simplified expression a + b
In other words, 16a + 5b - 10a + 2a - b - 7a - 3b is the same as a + b
Subtracting identical terms
3 - 3 = 0
45 - 45 = 0
12395 - 12395 = 0, etc
x - x = 0
2a - 2a = 0
100P - 100P = 0, etc
A quick question - what is the outcome of 10y - 6y - 4y?
You may have noticed that I have made no mention of positive and negative numbers and terms. We'll look at this tricky topic in a later article. For now, we'll keep things simple...
The order of addition and subtraction.
The order in which we carry out simple addition and subtraction is not important. 10 - 2 + 5 is the same as 10 + 5 - 2, or 5 - 2 + 10. Don't forget, though, that the operator and the following term stay together. 10 - 2 + 5 is NOT the same as 10 + 2 - 5.
Similarly, adding or subtracting letters works the same way. x - y + z is the same as z - y + x, and the same as x + z - y.
Combine the letters and numbers to form a simple algebraic term
1. m * n 2. p * q
3. a * b * c 4. 7 * t * s
5. 6 * x * y 6. 10 * g * h * f
7. h * j * 926 8. c * b * 1
9. g * 1 10. s * 5 * t * 4
What are the numerical coefficients of the following terms?
11. 12c 12. 2xy
13. b 14. pqr
15. 5126x 16. 7gh
17. m * n * 5 18. c * 16 * d
Simplify these expressions as much as possible by adding or subtracting like terms
19. 5a + 6a + 3b - a
20. 10x + 2y - x - y
21. p + q - p + q
22. 8x + 7y + 6z - 5z - 4y - 3x
23. 15g + 9h - h + 50g
24. x - x + y - y + 100z
Evaluate these expressions by simplifying, and substituting the values where given
25. 6a - a + 10g + 3a
(where a = 9 and g = 5)
26 20x +y -19x +10y -5y
(where x = 8 and y = 7)
27. 15p - 10p + 3q + p - q
(where p = 1, no value for q)
28. a + 2b + 3c + 4d
(where a, b, c, and d = 10)
29. 3s + 9t + 54u - 4t - 4u + 2s
(no values given)
30. e + 5f + 2e - f - 4f - 3e
(where e = 97,174 and f = 3,259)
19. 10a + 3b
20. 9x + y
22. 5x + 3y + z
23. 65g + 8h
27. 6 + 2q (or 2q + 6)
29. 5s + 5t + 50u