Unit 16 Section 1 : Fundamental Algebraic Skills

This section looks at some skills you will need as you start to learn about algebra.
It starts with some work on codes, then moves on to work with letters and formulae.

Encoding letters to other letters

Look at the code wheel below. Each letter on the outer ring can be encoded by looking at the corresponding letter on the inner ring. A letter can be decoded by finding it on the inner ring and looking at the corresponding letter on the outer ring.
 
Remember that the outer ring contains the uncoded letters and the inner ring contains the coded letters.

Example
Encoding the word FOX gives DMV, and decoding the word
NGE gives PIG. Check on the wheel that you can encode FOX
and decode NGE.

Practice Questions
Do the following encodings/decodings and then click on the button marked Click on this button below to see the correct answer to see whether you are correct.
(a) Encode the word MATHS using the code ring.

(b) Decode QMLGA using the code ring.

In algebra, we use letters to represent particular numbers which is a bit like using the code above.

Substituting numbers for letters

For this example, we need to give the letters a, b and c some values.
We will let a have the value 4, b have the value 7, and c have the value 3.
Normally we would write: a = 4, b = 7, c = 3.

If we see an expression which contains these letters, we need to replace the letters by their
corresponding numbers to find the value of the expression. Look at the examples below:
(a) Find the value of 6 + b:
Replacing b by 7 gives 6 + 7.
The answer is 13.
(b) Find the value of 2a + b:
Remember that 2a means 2 a.
Replacing a by 4 and b by 7 gives 24 + 7.

The answer is 15.
(c) Find the value of ab:
Remember that ab means a b.
Replacing a by 4 and b by 7 gives 4 7.

The answer is 28.
(d) Find the value of a(b - c):
You need to know that a(b - c) means a (b - c).
Replacing a by 4, b by 7, and c by 3 gives 4 (7 - 3).

The answer is 16.

Practice Questions
Calculate the value of each expression below and then Click on this button below to see the correct answer to see whether you are correct.
In this question, a = 2, b = 5 and c = 1.

(a) What is the value of b + 5?

(b) What is the value of 3a - c?

(c) What is the value of ab?

(d) What is the value of b(a + c)?

Simplifying expressions

Sometimes, we can simplify expressions by collecting like terms.
Like terms are parts of algebra which have the same letter (like 2x and 5x).
Look at the examples below:

(a) Simplify 2x + 5x.
2x is the same as x + x.
5x is the same as x + x + x + x + x.
Therefore, 2x + 5x must be the same as (x + x) + (x + x + x + x + x), which is 7x.

(b) Simplify 9y - 5y.
We have 9 lots of y and we are taking away 5 lots of y.
This will leave 4 lots of y, so 9y - 5y = 4y.

(c) Simplify 3a + 8b + 5a - 2b.
We can only collect together the terms which have the same letter (the like terms).
We have 3a and 5a which add together to make 8a.
We also have 8b and we are subtracting 2b, which will give 6b.
The 8a and 6b are added together to give 8a + 6b.

(a) Simplify 3p + 5q.
We can not combine these terms because they contain different letters.
Therefore, 3p + 5q is already simplified.

Practice Questions
Simplify each expression below, and then click Click on this button below to see the correct answer to see whether you are correct.

(a) Simplify 2x + 4x.

(b) Simplify 5p + 7q - 3p + 2q.

(c) Simplify y + 8y - 5y.

(d) Simplify 3t + 4s.

Writing formulae for particular situations

Look at the rectangle below:
We want to work out formulae for the area and perimeter of the rectangle.
To find the area, we multiply the length (x) by the width (y).
This is x y which is the same as xy
The formula is Area = xy
To find the perimeter, we add up the lengths.
This is x + y + x + y which is the same as 2x + 2y
The formula is Perimeter = 2x + 2y

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
Use the code wheel below to answer the two questions. The inner letters are the coded versions of the outer letters.
Give your answers to this question in CAPITAL letters.

(a) Work out the coded version of the message:

MEET ME AT HOME

(b) Decode the message:

MTCP RM WMS

Question 2
Use the code wheel below to answer the two questions. The inner letters are the coded versions of the outer letters.
Give your answers to this question in CAPITAL letters.

(a) Code the message:

GONE FISHING

(b) Decode the message:

TUST RUHQ

Question 3
If a = 2, b = 6, c = 10 and d = 3, calculate the value of:
(a) a + b (b) c - b
(c) d + 7 (d) 3a + d
(e) 4a (f) ad
(g) 3b (h) 2c
(i) 3c - b (j) 6a + b
(k) 3a + 2b (l) 4a - d

Question 4
If a = 3, b = -1, c = 2 and d = -4, calculate the value of:
(a) a - b (b) a + d
(c) b + d (d) b - d
(e) 3d (f) a + b
(g) c - d (h) 2c + d
(i) 3a - d (j) 2d + 3c
(k) 4a - 2d (l) 5a + 3d

Question 5
If a = 7, b = 5, c = -3 and d = 4, calculate the value of:
(a) 2(a + b) (b) 4(a - b)
(c) 6(a - d) (d) 2(a + c)
(e) 5(b - c) (f) 5(d - c)
(g) a(b + c) (h) d(b + a)
(i) c(b - a) (j) a(2b - c)
(k) d(2a - 3b) (l) c(d - 2)

Question 6
Use the formula s = (u + v)t to find s, when u = 10, v = 20 and t=4.

The value of s is

Question 7
Use the formula v = u + at to find v, when u = 20, a = -2 and t = 7.

The value of v is

Question 8
Simplify the expressions below, where possible.
(a) 2a + 3a
(b) 5b + 8b
(c) 6c - 4c
(d) 5d + 4d + 7d
(e) 6e + 9e - 5e
(f) 8f + 6f - 13f
(g) 9g + 7g - 8g - 2g - 6g
(h) 5p + 2h
(i) 3a + 4b - 2a
(j) 6x + 3y - 2x - y
(k) 8t - 6t + 7s - 2s
(l) 11m + 3n - 5p + 2q - 2n + 9q - 8m + 14p

Question 9
Complete the formula for the perimeter of each of the shapes below.
(a) Perimeter =
(b) Perimeter =
(c) Perimeter =
(d) Perimeter =
(e) Perimeter =
(f) Perimeter =

Question 10
Sam asks her friend to think of a number, n.
She then asks her friend to multiply her number by 2 and then add 5.

Complete this formula for the number her friend ends up with.
Final number =

Question 11
A removal firm makes a fixed charge of 50, plus 2 for every mile travelled.

Complete the formula for the cost (in ) of a removal when travelling x miles.
Cost =


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Produced by A.J.Reynolds January 2001