MATHS - ALGEBRA
Introducing Algebraic Expressions
In maths, we use symbols to represent numbers, operations and formulae. It is important to understand how these are arranged.
It is useful to be able to explain mathematical ideas using words. However, it is equally important to be able to use symbols - such as numbers (e.g. 5), letters (e.g. x) and signs (e.g. +).
We often use x to represent some number in a mathematical calculation. Sometimes, we refer the letter x as a variable or a pronumeral.
EXAMPLES
x + 4 means a number with 4 added on.
x - 20 means 20 less than x, or the result of taking away 20 from a number.
MULTIPLICATION & DIVISION
The two symbols x and ➗ are not used very often in algebra.
We leave out the multiplication sign.
EXAMPLE
5 lots of x, or 5 x X, is written as 5x.
EXERCISE 1.01
- Match each algebra expression with an english phrase from the box.
- x + 3
- x - 2
- 5x
- x + 5
- 2x
- 3x
- x - 1
2. Write down algebra expressions for the english expression.
- A number with 10 added to it.
- A number with 15 subtracted from it.
- A number multiplied by 4.
- A number multiplied by 7.
3. Write down the english expression for each algebra expression.
- 6x B. x + 9
C. x - 12 D. 5x
4. For each english expression, write down an algebra expression chosen from the box.
- Ten times a number.
- A number divided by 10.
- Ten divided by a number.
- Ten more than a number.
- Ten less than a number.
5. Write down these expressions without using multiplication or division signs.
- 6x B. x ÷ 6 C. 20 x X D. x ÷ 30
E. p x 9 F. 12 x y G. 100 ÷ x H. p ÷ 9
6. Write an equivalent algebraic expression for each of these.
- x take away 7.
- Six more than x.
- Twelve lots of x.
- Ten less than x.
- The number of cents in x dollars.
- The number of days in x weeks.
- The sum of p and q.
7. Show the meanings of these expressions by including multiplication signs.
- 4p B. 6p - 1
C. pq D. 2 + 3p
8. Rewrite these expressions, showing all the implied multiplication and division signs.
- 12 x B. x/6
C. cd D. 7/y
Substitution [replace the variable (letter or symbol) with a number to work out the value]
If x = 7, then x + 3 = 10.
But if x = 24, then x + 3 = 27.
The number represented by x + 3 depends on the value of x.
EXERCISE 1.02
- The value of x is 8. Calculate each of these expressions.
- x + 2 = B. x + 10 = C. x + 27 = D. x + 6 =
E. x + 8 = F. x + 1 = G. 2 x X = H. 4 x X =
- 7 x X = J. 3x = K. 10x = L. 9x
M. x ÷ 4 = N. x/2 = O. x/8 = P. 48/x =
2. A car factory uses a rule to decide how many tyres it needs to purchase when it is assembling cars. The rule is:
The number of tyres = 5 x the number of cars
The manager writes down this rule as:
Number tyres = 5x
- What does x stand for in this expression?
- The factory is assembling 60 cars. How many tyres does it need to purchase?
- What is the value of 5x when x = 40?
EXERCISE 1.03
- Here a = 4, b = 5, c = 13 and d = 28. Calculate the value of each of these expressions.
- a + b B. a + c + d C. d + b
D. b - q E. c - b F. d - a
2. An airline provides two types of dinner on a flight - chicken and fish. The total number of meals served on a plane can be worked out using the rule c + f. What is the value of c + f when:
- c = 41 and f = 35?
- c = 56 and f = 29?
3. Sounds Unlimited use this rule to work out the number of CD’s they sell each week: Number sold = number in shop at start of week - number in shop at end of week.
- Explain what x stands for in this rule.
- Explain what y stands for in this rule.
- Calculate the value x - y when x = 20 and y = 7.
- Explain in words what you have worked out in part c.
SIMPLIFYING ALGEBRAIC EXPRESSIONS
There are two ways we can show multiplication:
- with at ‘times’ symbols (x),
- by placing the letters, or number and letter, next to each other.
Example
Each of these are equivalent.
Using the x symbol
|
Simplified
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2 x X
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2x
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p x q
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pq
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Here are the rules writing multiplication expressions as simplified as possible.
Rule
|
Example
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Any number in the expression are multiplied
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4 x 2x = 8x
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If there is more than one letter, they are written in alphabetical order
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3d x 5c = 15cd
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Numbers are placed in front of letter when multiplying
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p x 2q = 2pq
|
Exercise 1.04
Simplify these expressions.
- 4 x 3x 8. 4c x2d 15. 4d x 2c x e
- 2 x 5x 9. 2g x 3f 16. 3 x 2a x 5
- a x c 10. p x 2a 17. 1 x 10q x 1
- 3 x p 11. 2a x 4b 18. 2 x 4p x 2q x r
- 1 x d 12. q x 2p x 3r 19. 4a x b x 2c
- 7 x 2x 13. f x 2d x 5e 20. q x r x p
- b x a 14. 3o x q x 2p
Expressions with like terms
Like terms are those that have the same combination of letters. For example 3ab, -5ba and ab are all like terms. Other examples of like terms are: (a) x2, 3x2, -5x2 (b) ac2e, -3ac2e, 5ac2e
Unlike terms have different letter combinations. For example (a) ab and bc are unlike terms. (b) x and 2x are unlike terms because the powers of x are different.
Complete the Statements
- A algebraic ___________ is made up of __________ and operators such as +, -, x, ÷ and ().
- A ______ is made up of numbers and letter symbols but not operators.
For Example,
- 3a + 4b - a is an ______ b. 3a, 4b and 5 are _______ in the expressions.
c. 3a and a are called ______ because the both contain a number and the letter symbol a.
Choose your answer from: expression, term,
like term
Like terms can be added and subtracted
Examples
- 5 dogs + 7 dogs = 12 dogs
5d + 7d = 12d
2. 14 pigs - 6 pigs = 8 pigs
14p - 6p = 8p
3. 8 mangoes + 5 apples - 3 mangoes - 2 apples = 5 mangoes + 3 apples
8m + 5a - 3m - 2a = 5m + 3a
4. 6x2y + 5x2y = 11x2y
5. 3x + 5 + 6x - 2 + 7 + y = 9x + 10 + 9x + y
