Algebra (Basics)
Question One
Nathalie du Toit swims 12 lengths every day to get fit. How many lengths does she
swim in:
1.1
1.2
1.3
1.4
3 days
5 days
10 days
x days
1.5
1.6
1.7
1.8
2 weeks
3 weeks
10 weeks
p weeks
In algebra we write an “x” as x (italics) or x or
Question Two
The entrance fee to a soccer match is R200 for adults and R100 for children. Work
out the total entrance fee for:
2.1
2.2
2.3
2 adults and 1 child
1 adult and 3 children
3 adults and 2 children
2.4
2.5
2.6
5 adults and 7 children
2 adults and 4 children
x adults and y children
Question Three
Write an expression for each of the following:
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
the sum of the two numbers 12 and 7
the sum of two numbers a and b
the difference between the two numbers 6 and 2
the difference between two numbers m and n
the product of 5 and 4
the product of p and q
the quotient of 13 and 6
the quotient of r and t
the sum of 9 and twice 8
the sum of x and twice y
half of 14 subtracted from double 6
half of p subtracted from double d
the sum of three consecutive numbers starting with 6
the sum of three consecutive numbers starting with m
Question Four
3 + 3 + 3 + 3 can be written as 4 x 3
Write the following in another form:
4.1
4.2
4.3
4.4
4.5
7+7+7+7+7
x + x + x+ x
b+b+b+b+b+b+b
(b + b + b) + (b + b)
m + m + m + . . . to 26 terms
Key Facts
3p means 3 x p
b means 1 x b
c2 = c x c
we write 5ab not ab5 and not 5ba
Question Five
Colin is x years old. Give an expression for each of the following in terms of x .
5.1
5.2
5.3
How old was he last year?
How old will he be in 5 year’s time?
How old was he y years ago?
5.4
5.5
How old will he be in z year’s time?
How old will he be if you double his age?
Question Six
A journey is travelled at a speed of 100 km/h.
6.1
6.2
6.3
What is the distance travelled in 1 hour?
What is the distance travelled in 5 hours?
What is the distance travelled in x hours?
Question Seven
Think of a number.
Add 7.
Double your answer.
Subtract 4.
Divide by 2.
Subtract the number you started with.
7.1
7.2
Write down the answer you get.
Try to write an expression showing what you did.
Let the number you started with be x .
Question Eight
Simplify:
8.1
8.2
8.3
8.4
8.5
8.6
8.7
3x a
5xb
cx4
a x b
1 x t
m x0
d x b
8.16
8.17
8.18
8.19
8.20
8.21
8.22
3x2x4xp
(3 + 2) x a
b x (4 + 3)
p x q x (2 + 5)
a x (3 + 1) x b
c x d x ax b
x x2x y x5
8.8
8.9
ax b x c
2 x 3 a x 5b
8.23
8.24
a b x cd
8.10 3 x 4 x y
8.25
2 x 3 x x
8.11 2 x 3 x x x y
8.26
3 x x x 4
8.12 2 x x x y
8.27
 x x ym
8.13 c x d x 2 x 2
8.14 x x 0 x y
8.28
8.29
1xpxqxr
2 a x 3b x 4c
8.15
8.30
q x p x (2 + 3)
x x yx2x0
xyz x w
Question Nine
True or false? If false, correct them:
9.1
9.2
5d + 8d = 13d
2x+ 4 y+ 3x = 5x + 4 y
9.5
9.6
12 x + 12 x + 12 x = 36p
7d  6d = d
9.3
9.4
15t  6t = 9
6x 6 = x
9.7
9.8
12 x  9 x =  x + 4 x
23 x 2 + 7 x 2 = 30 x 4
Question Ten
The formula that converts degrees Fahrenheit to degrees Celsius is:
C
=
5 (F – 32)
9
Calculate the temperature in degrees Celsius in each of the following cases.
Calculators may be used.
10.1
10.2
10.3
10.4
32° F
122° F
302° F
59° F
10.5
10.6
5° F
0° F
10.7 If water boils at 100° C at sea level, what is this in ° F?
Substitution
Example:
=
=
=
Find the value of b2 + bc  2c  c2 when b = 2 and c = 3
(2)2 + (2)(3)  2 (3)2 – (3)2
4  6  2 (9)  (9)
4  6  18  9
29
Question Eleven
Find the values of each of the following expressions when a = 1 ; b = 2 and c = 2
11.1
11.2
11.3
11.4
11.5
11.6
ab
abc
a+b+c
b+c
abc2
2ac
11.7
11.8
11.9
11.10
11.11
11.12
a+b+c
ab
cb
a2
b2
(a + b) (a + c)
Question Twelve
Find the values of the following expressions when
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
ab
abc
ad
a2 + d2
ab  bd
a2  b2
a2
ab + 6
da
ad  c
12.11
12.12
12.13
12.14
12.15
12.16
12.17
12.18
12.19
12.20
a = 2 b = 3 c = 1 d = 0
cd  ab
b+c
abcd
bc  a  d
acd  abc
d2  b2 + c2
ab + bc + cd
ac2 + bd2
(abc)2
ac2 + b2 + c
Question Thirteen
Find the values of the following when x = 2;
y = 4 and
z = 3
13.1
xy  yz  xz
13.5
5(3x  2 y  z )
13.2
x y z
13.6
( x  y)( y  z )( x  z )
13.3
4x  3 y  2z
13.4
x( x  y )  y ( x  z )
2
2
2
Question Fourteen
If a = 2; b = 4 and c = +5 find the values of:
ab  bc
a
2a  3b  4c
b
14.1
(a  b)2
14.5
14.2
(a  b)2
14.6
14.3
(b  c)a
14.7
a 2b  b 2 c  c 2 a
14.4
ba
14.8
(2a  3b)a
Terms, Coefficients, Variables and Constants
 Terms are separated by a + or a –
Example: 5 x  3 y  2
 2x  3 y   3 y  2z



3 terms
2 terms
Variables are the letters
Coefficients are the numbers that sit in front of the variables
Constants are any numbers without variables
Example: In the expression 3p + 2bc  4
-
There are 3 terms
There are no like terms
4 is called a constant term
Question Fifteen
How many terms are there in each of the following algebraic expressions?
15.1
5 x  4 y  3z
15.7
( x  1)  ( x  3)
15.2
2xaxbxc
a  b 2a

5
3
3b
 2a
5
1
5a  3b 
2a
15.8
3a  4b  2c  abc
15.9
3a  2b  2c
abc
( x  1)( x  2)
15.12
15.3
15.4
15.5
15.6
15.10 (a  b)  (c  d )
15.11 2a  (c  d )
ab
cd
Question Sixteen
Write down the coefficient of the variable x in each of the following algebraic
expressions:
16.1
16.2
3x
3x + 5
16.6
16.7
3x2 + x
7x  5y + 1
1  2x
2x x x5
16.3
16.4
5 + 3x
2 x + yz
16.8
16.9
16.5
x
16.10 2 ( x + 5)
Question Seventeen
From the expression 2 x + 5, name:
17.1
17.2
the constant
the coefficient of x
17.3
17.4
the variable
the number of terms
Question Eighteen
9 x2  7 x  3  y 2  13 y
Copy and complete the sentences below using the following list:
Given:
9 ; y 2 ; 7 x ; variable, constant, term
18.1
18.2
18.3
18.4
x is a _______
3 is the ____
___ is the coefficient of x 2
1 is the coefficient of _____
Question Nineteen
Write an algebraic expression for each of the following:
19.1
A man is p years old. How old was he m years ago?
19.2
I travel at a speed of d km/hr. How many kms do I travel in y hours?
19.3
10 apples cost g rand. How much will one apple cost?
19.4
James is 15 years old. How old will he be in m years’ time?
19.5
At a burger bar, the tables can seat 6 people. How many people can be
seated at m tables?
Question Twenty
Translate into algebraic language:
20.1
the product of m and n is greater than p
20.2
if g is divided by h the result is b
20.3
the difference between x and y is 5
20.4
the difference between e and f is 10
( x is bigger than y )
(f is bigger than e)
20.5
the sum of c and d, divided by 2, is less than 9
20.6
the square of m minus the square of n is greater than twice m
20.7
the sum of the squares of p and q is equal to 52
20.8
the square of the sum of a and b is 100
20.9
the sum of b and half of c is equal to 18
20.10 five times p, decreased by 3, is equal to d
Question Twenty One
Something fun to make you think 
21.1
If there are 10 posts in a straight line, how many spaces between them?
21.2
If there are n posts in a straight line, how many spaces between them?
21.3
If there are b spaces between posts in a straight line, how many posts are
there?
The Distributive Law
(Basic)
Question One
Investigate the following:
1.1
If a  2 ; b  3 and c  4 , evaluate the following:
a(b  c)
______________________________________________________________
ab  ac
______________________________________________________________
1.2
If a  3 ; b  1 and c  4 , evaluate the following:
a(b  c)
______________________________________________________________
ab  ac
______________________________________________________________
1.3
If a  2 ; b  3 and c  1 , evaluate the following:
a(b  c)
______________________________________________________________
ab  ac
______________________________________________________________
a(b  c) =
___________________________regardless of the values of a, b and c
a(b  c) =
___________________________regardless of the values of a, b and c
Distributive Law
 The Distributive Law means that you get the same answer when you
multiply a number by a group of numbers added together as when you
do each multiplication separately.
 We ‘Rainbow’ the term outside the bracket to EVERYTHING inside the
bracket
2a  b
 2a  2b
Polynomials
 An expression with 1 term is called a monomial
 An expression with 2 terms is called a binomial
 An expression with 3 terms is called a trinomial
Question Two
Categorise each of the following polynomials correctly:
2.1
2x 1
2.2
3x 2  2 x  3
2.3
2x 4
2.4
x  4  5 x3
Question Three
Simplify the following using the distributive law.
3.1
3.2
3.3
2( x  y)
3( x  2)
x(2  y)
3.4
3.5
3.6
a(b  c)
2( x  y)
3( x  2)
3.7
3.8
3.9
x(2  y)
a(b  c)
3(2a  4b)
Question Four
Simplify the following:
4.1
3( x  4)
4.6
(a  1)(4)
4.2
( x  4)(3)
4.7
6(a  1)  4
4.3
( x  4)  3
4.8
( x 2  3)2
4.4
2( x  1)  2
4.9
( x 2  3)  2
4.5
2( x  1)(2)
4.10
3x  ( x  2)
“Arithmetic is being able to count up to twenty without taking off your shoes."
– Mortimer Mouse (later called Mickey Mouse)
Like and Unlike Terms
Like Terms
 Like terms have the exact same variable and exponent
Example:
x , 3x ,  7x
LIKE terms
3 x2 y ,  5 x2 y ,14 yx2
ab , a2b2 , a2b , ab2
LIKE terms
UNLIKE terms
 Like terms can be added and subtracted, unlike terms MAY NOT be
added nor subtracted
Example:
1) x  3 x  7 x  3 x
2) 3 x2 y  5 x2 y  14 yx2  12x2 y
Question One
Simplify the following fully:
1.1
1.2
2x a
3 x ab
1.15
1.16
ab  ab  ab  …
xy  xy  xy  …
1.3
1.4
3x b
x x y
1.17
1.18
3a x 2b x 4c
3x4x2xa
1.5
1.6
1.7
1x x x2
2x a x3x b
4x x x y x2
1.19
1.20
1.21
8r  r
18 a + 3 a  2 a
13 ab  ab + 4 ab
1.8
1.9
1.10
2x a x b x c x5
xxxxxx
x x5
1.22
1.23
1.24
14 x + 5 x  4 x  x
abc + 4 abc  5 abc
2 zxy + 9 zyx  4 xyz
1.11
1.12
1.13
mmm . . .
xxx . . . . .
2 a + 2 a + 2 a + ..
(10 terms)
(20 terms)
(10 terms)
1.25
1.26
1.27
4d  5d
5b + 2b
6 f  7 f
1.14
3r + 3r + 3r + …..
(20 terms)
1.38
5kmn  3knm  12mkn
( c terms)
( z terms)
Question Two
Simplify each of the following fully. Remember to use BODMAS! You may insert
brackets where necessary to help you.
2.1
2.2
2.3
2.4
2.5
2ax 4 + 3ax 3
4m x 3 + m x 2
5b x 3 + 5b x 4
2c + 4c x 3
5t + 3 x 2t
2.15
2.16
2.17
2.18
2.19
10m  4m 5m
12 a  5 a x 3
15t  12t x 1
9xb3xbbx2
4f x23f x3
2.6
2.7
7k + k x 4
3f x1+3 f x1
2.20
2.21
a x b + 2a x 3b + 3a x b
4g x 2 f  3g x 6 f
2.8
2.9
2.10
2.11
2.12
2.13
2.14
4ax 3 + 3a
9k + 4 x 4k
2d x 1 + 3d x 1
3e x 0 + 2e x 1
5x x +0
5x x x0
2.22
2.23
2.24
2.25
2.26
2.27
p + 3 x p + 4 x p  2p x 2
3d x 4  0 x d
3w x 2 – 4 x 2w
5r2x6r
4fx26fx2
7y  3y x 2
dx2+3xd+dx4
Question Three
Choose the term in each list which is unlike the rest of the terms:
3.1
3.2
3.3
a ; 2a ; b ; 3a
x ; 5 x ; 5y ; 3 x
ab ; ba ; 2 ba ; 3 a
3.4
5 xyz ; 3 xyz ; 2 zyx ; 5 xy
3.5
3.6
7y ; 7 ; 3y ; 4y
6 x ; 5 x ; 5 x2 ; 9 x
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
5a + a  2a
7 a + 2 a
ab + ba
4 x + 4y + 2
ab + bc
13mnp + 4pmn
a + c + ac
2y + 5y  3y
9a  a
Question Four
Simplify as far as possible:
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
x + 2x
4 f + 7 f
x + 2y
a + 3b
6b + 12b
7 x  7
10 a  a
9r  9s
ab  ac
Question Five
Simplify:
5.1
3x+ 4 y+ 2x + y
5.13
5x+ 3 y+ 2x+ 2 y
5.2
5.3
5.4
5.5
5.6
5.7
5 a + 2b + 3b  4 a
4c + 2d + c + 5d
x + 2 + 2x + 3
7h + 2 + 3h + 5
5  2s  3 + 7s
xy + 4 xz + 2 xy + 3 xz
5.14
5.15
5.16
5.17
5.18
5.19
7 a  4b  2b + 3 a
9y + 7z + 3y + 2z
7w + 3 x + 4w + 2 x
7b  9c + 3b + 2c
6 ab + 2 ac + 3 ab + 2 ac
10 xy + 4 xz + 9 xy + 3 xz
5.8
5.9
2pq + 3rq + 5pq + 9qr
5 xy + 3 xz + 9 xz + 2 xy
5.21
5.20 5 ab + 6 ac + 5 ab + 6 ac
9ef + 7fg + 2fe + 3gf
5.10
5.11
3
5.12
2 ab + 3 ac + 4 ab + 9 ac
10mn + 11mn + 2m + 3n
5.22
5.23
2 a 3b5c + 5 a + 2b + 4c
2y + 3y + 5 + 3y + y + 7 + 2y +
5 abc + 3 bc + 2 bac + bc
Question Six
Simplify:
6.1
6.2
6.3
6.4
6.5
6.6
6.7
12 ab + 13 ab
15 a 2 + (15 a 2 )
3 x 3 + (+15 x 3 )
(12 ab ) + (30 ab )
(8 a ) + (3 a )  (4 a )
(10 ab ) + (+2 ab ) + (5 ab )
5 x  (+2 x )
6.8
6.9
6.10
5 x  (2 x )
(5 x ) (+2 x )
7 a  (3 a )
6.17
6.11 7 a 2  (11 a 2 )
16 ab  6 ab
bc  (5bc)
4m  12m
6.15 4 x  (2 x )  3 x
6.16 7y  (2y)  y
3 xy  (2 xy )  5 xy
6.18
9 x 2  3 x 2  ( x 2 )
6.12
6.13
6.14
Multiplication of Algebraic Expressions
When multiplying…
 Multiply the signs
 Multiply the numbers
 Multiply the variables
Question Seven
Simplify:
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
(2 a x b) + (3 a x b) + ( a x b)
(4 a x 2b) + (2b x c) + ( a x b) + (b x c)
(2 x x  2y) + (2y x 3z) + (3z x 4y) + ( x x y)
(3 a x 2) + (3b x 4) + (5 a x 3) + (6b x 1)
( ab x c ) + (2 bc x d ) + ( c x 2 ab ) + (3 bc x 4 d )
(4y x 2) + (2 x 5) + (3 x 4) + (4 x 4y)
(5b x d) + (4 a x 3c) + (7 a x 2c) + (3 a x 2c)
(6m x n) + (3m x n) + (4n x p) + (2n x p)
(3 a x 2) + (4b x 2) + ( a x 2)  (b x 2)
( x x y)  (8y x 2z) + (2 x x 2y)  (4y x 3z)
More Complicated Addition of like terms
Example:
Add: 5 x + 4y + 3; 2 x + y + 1; 6 x + 3y + 2
Question Eight
Add the following expressions.
8.1
6 x + 3 xy + 4 y ; 3 x + 2 xy + y ; 2 x + xy + 5 y
8.2
8.3
8.4
8.5
8.6
8.7
7 a + 3b + c ; 4 a + 2c + b ; 6 a + 5c + 3b
2 ab + 3 bc + 2 cd ; 4 bc + cd + 5 ab ; 6 bc + 3 ab + cd
6 a + 3b ; 2b + 4c ; 5 a + 5c
7 x2 + 4 x + 2 ; 3 x2 + 5 x + 5 ; x2  2 x  3
7p + 4q  r ; 3q + 5r  10p ; 3r  p + 4q
5 x 2 + 2 y 2 ; 3 y 2 + 2 z 2 ; 6 z 2 + 5 x 2
8.8
4 a 2 + a 3  5 a ; 2 a 2 + 3  a ; 6 a + a 2  7
More Complicated Subtraction
Example:
1) Subtract 2d  2e  2df from 10d  10e  6df
2) From 4b  2c  10d subtract 7b  5c  2d
Question Nine
9.1
9.2
9.3
9.4
9.5
From 5a + 3b + 10c subtract a + 2b + 4c
From 6b + 4c + 10 subtract 8 + 2c + 3b
Subtract 8y + 2z + 5 from 10y + 10z + 7
How much larger is 11 a + 3 b + 5 than 3 a + 2 b + 2?
How much smaller is 3 x + 2 y + 3 than 7 x + 8 y + 10?
9.6
9.7
9.8
Subtract ab from a
Subtract 10 a - 2 b from 6 a  4 b
Find the sum of 3 x  y and  x  5 y
9.9
From 5 x 2  2 x subtract 6 x 2  20 x
9.10 From 0 subtract 7 x  7 y
9.11 Subtract 5 a  4 b + 10 c from 6 b  3 c + 4 a
9.12 Subtract 2c  3d from 4e  5f
9.13 From 15 x 3 + 4 x 2  8 x subtract 6 x 3  9 x + 6