Basic Mathematics Form One Notes – Algebra and Equations
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‘An algebraic expression — is a collection of numbers, variables, operators and grouping
symbols, Variables – are letters used to represent one or more numbers
Algebraic Operations
Symbols to form Algebraic Expressions
Use symbols to form algebraic expressions
The parts of an expression collected together are called terms
Example
+ x+2x—are called like terms because they have the same variables
+ 5x+9y—are called unlike terms because they have different variables
‘An algebraic expression can be evaluated by replacing or substituting the numbers in the
variables
Example 1
Evaluate the expressions below, given that x = 2 and y= 3
a1
(a) Sx-y
(b) 3(x + 2)
(c) ae +2y
Solution
(a) 5x-y =5(2)-3=10-3=7
(b) 3(@@ +. 2) = 3(2 + 2) = 3(4) = 12
(©) $x + 2y =(2) +208) =24+6 = 62
Example 2
Evaluate the expressions below, given that m= 1 and n= -2
2
3
(@) oa
m+1
m
5)
(b)
(c) m-n
Solution
3 3.3 1
(a) = =2=1+
m+i 141 2 2
m 1 dd
(b) = =-+-=!
i-n i–2 142 3
(c) m—n=1—- -2=14+2=3
‘An expression can also be made from word problems by using letters and numbers
Example 3
‘A rectangle is 5 cm long and w cm wide. What is its area?
Solution
Let the area be A.
Then
‘A= length widith
A=Swem
Simplifying Algebraic Expressions
Simplify algebraic expressions
Algebraic expressions can be simplified by addition, subtraction, multiplication and division
83
Addition and subtraction of algebraic expression is done by adding or subtracting the
coefficients of the like terms or letters
Coefficient of the letter — is the number multiplying the letter
Multiplication and division of algebraic expression is done on the coefficients of both like and
unlike terms ot letters
Example 4
‘Simplify the expressions below
(a) 4a+3b+2a+b
(b) 6n+3m—2n—2m
1 1
(c) =s-t+3s—=t
2 4
(d) Smn—3nm
Solution
(@) 4a +3b+2a+b
Collect like terms together
4a+3b+2a+b=4a+2a+3b+b
= 6a+4b
(b) 6n + 3m —2n—2m
Collect like terms together
6n + 3m — 2n— 2m = 6n—2n+ 3m—2m
=4nt+m
1 1
(© 4s—t4+3s—4e
Collect like terms together
L t+3: L t= 4 +3: t i 3
Pia ST gta gst eso ig
=3ts-1tt
a4
(4) 5mn — 3nm = 5mn — 3mn = 2mn
a
Equations with One Unknown
‘An equation — is a statement that two expressions are equal
‘An Equation with One Unknown
Solve an equation with one unknown
‘An equation can have one variable (unknown) on one side or two variables on both sides.
When you shift a number or term from one side of equation to another, its sign changes
+ Ifitis positive, it becomes negative
+ Ifitis negative, it becomes positive
Example 5
Solve the following equations
(a) x+3=5
1
(b) x-—2-=8
4
(c) 3x+4=-3
(d) 5x+11=18
(e) 5-—3x=24
Solution
85
(a) x+3=5
x=5-3
x=2
1
(b) x-27=8
ten
air
=8+ a
plat
-8,9_ 3249 _41_ 1456
ata 4 a tO
x= 10.25
(c) 3x+4=-3
3x=-3-4
3x=-7
7
x=-3=-233 , x=-233
3
(d) 5x+11=18
5x = 18-11
Sx =7
Te
aged
x= 14
(e) 5-3x =24
-3x = 24-5
-3x = 19
19 19
xongo 7 7633
x = 633
An Equation from Word Problems
Form and solve an equation from word problems
‘Some word problems can be solved by using equations as shown in the below examples.
Example 6
86
Naomi is 5 years young than Mariana. The total of their ages 33 years. How old is Mariana?
Solution
Let the age of Mariana be x
Naomi= x —5
Then x+(x-5) = 33
x+x-5=33
2x =33+5
2x = 38
38 49
x=>=
2
x=19
Mariana is 19 years
Equations with Two Unknowns
Simultaneous Equations
‘Solve simultaneous equations
‘Simultaneous equations ~ are groups of equations containing multiple variables
Example 7
Examples of simultaneous equation
a7
5 xty=1 sy ME 25213 og [DAO
«) Pai =, w fA Sam 34
A simultaneous equation can be solved by using two methods:
. Elimination method
. Substitution method
ELIMINATION METHOD
STEPS
° Choose a variable to eliminatee.g x ory
. Make sure that the letter to be eliminated has the same coefficient in both equations and if
not, multiply the equations with appropriate numbers that will give the letter to be eliminated the
same coefficient in both equations
i 2-8y— 2 aininatert it has th i
(i) 2x+3y = 4 liminate y because it has the same coqffi
a t= by=el — a
(ii) _ eliminate x because it has the same cofffi
x+y=2
ave 5x+4y=2
(iii) y
3x+y=3
* To eliminate x, multiply equation (i) by 3 and (if) t
ie 3 Sxt4y=2
5 3x+y=3
a = #127 =6:
15x + 5y = 15
a8
= To eliminate y, multiply equation (i) by 1 and (ii) by 4
ie 1 Sx+4y=2
4 3x+y=3
= { 5x+4y=2
12x + 4y = 12
+ If the signs of the leter to be eliminated are the same, subtract the equations
. If the signs of the letter to be eliminated are different, add the equations
Example 8
Solve the following simultaneous equations by elimination method
3x+y=9
(@ es -y=7
3x —2y = 13
() f 3x+2y=1
5r-g=14
© i. +3g =15
7x + 6y =8
@) – —-3y=7
Solution
a Eliminate y
2
a fee PY scsssease (DO)
5x — y = 7 (ti)
8x = 16
_16_,
ra
x=2
To find y put x = 2 in either equation (i) or (i)
From equation (i)
3x+y=9
3(2)+y =9
6+y=9
y=9-6
Yeas
e x=2, y=3
(b)Eliminate x
J) & = Zy = 13… 00 (1)
3x + 2y =1……. (i)
—4y = 12
y =-12
— 12_ 5
: Jaci Pas
a3
90
In order to find y, put x = 2 in either equation (i) or (ii)
From equation (ii)
3x+2y=1
3x + 2(-3)=1
3x-6=1
3x=1+6
3x=7
_7
aa 43
2!
x= 23
wx=22, y=-3
3
(©) Given
Bt
{ 5r—g =14…….. (0)
4r + 3g =15………(ii)
Eliminate g
3/5r-—g =14
1|4r+3g =15
i —3g = 42
+ | ar 43g =15
19r =57
_ Tg
T= 59>
r=3
To find g put r= 3 in either equation (i) or (i)
From equation (i)
5Sr—g=14
5(3)-g=14
15-g=14
—g=14-15
=—g=—i
g=1
2 r=3, g=1
(a) Given
2
ie + 6y = 8……. (i)
2x — BY = 7 oe vee (ii)
Eliminate x
2|7x + 6y =8
7\2x—3y=7
lige 12y = 16
14x — 21y = 49
33y = —33
_ 33 _ i
2k ABS
y=]
To find x, put y=- 1 in either equation(i) or (ii)
From equation (ii)
2x-—3y=7
2x —3(-1)=7
2x+3=7
2x=7-3
2x=4
5
x=o=
2 x=2, y=-1
BY SUBSTITUTION
STEPS
+ Make the subject one letter in one of the two equation given
3
x—3y =B……(i)
bid a + 3y =4 anes (ii)
Make x the subject from equation (i)
x=8 + 3y……. (iii)
+ Substitute the letter in the remaining equation and proceed as in case of elimination
2x+3y=4
2(8 + 3y) + 3y =4
16+ 6y+ 3y=4
16+ 9y=4
9y = 4-16
9y = —-12
_ 12, «4
—_e 3
From (iii)
x=8+3y
=2843/ i)
“ 3
x=8-4=4
4
« x=4 or —=
3
Example 9
Solve the following simultaneous equations by substitution method
2x+y=17
(a) { x+y=4
Solution
94
2x + Y = 17 ween (i)
{) – ¥Y = 4 essences Gi)
From (ii)
X= 4 = ce enssssee (iii)
Substitute (iii) into (i)
ax+y=17
2(4-y)+y=17
8—2y+y=17
8-y=17
—y=17-8
-y=9
y=-9
From (iii)
x=4-y
x=4—-9
x=44+9=13
2 x=13, y=-9
Linear Simultaneous Equations from Practical Situations
Solve linear simultaneous equations from practical situations
Simultaneous equations can be used to solve problems in real life involving two variables
Example 10
If 3 Mathematics books and 4 English books weighs 24 kg and 5 Mathematics books and 2
English books weighs 20 kg, find the weight of one Mathematics book and one English book
Solution
95
Let the weight of one Mathematics book = x and
Let the weight of one English book = y
Then
(= HAY = 24 wee i)
5x + 2Y = 20 sr see ne (Hf)
Eliminate y
2|3x + 4y = 24
4|5x + 2y = 20
_| 6x + By = 48
20x + By = 80
—14x = —32
=
x=age2
x = 2.29
To find y, put x = 2.29 in either equation (i) or (ii)
From equation()
96
3x + 4y = 24
3(2.29) + 4y = 24
6.87 + 4y = 14
4y = 24-687
4y = 17.13
17.13 #2p
: aay Waleed
2 x=2.29, y=4.28
Inequalities
‘An inequality — is a mathematical statement containing two expressions which are not equal. One
expression may be less or greater than the other.The expressions are connected by the inequality
symbols<,>,< or: Where< = less than,> = greater than,< = less or equal and > = greater or equal
Linear Inequalities with One Unknown
Solve linear inequalities in one unknown
‘An inequality can be solved by collecting like terms on one side.Addition and subtraction of the
terms in the inequality does not change the direction of the inequality Multiplication and division
of the sides of the inequality by a positive number does not change the direction of the
inequality. But multiplication and division of the sides of the inequality by a negative number
changes the direction of the inequality
Example 11
Solve the following inequalities
7
(a) x+8>15 (b) 7—2x<11 () fx-32-2+x (d) Br
@) S220 (f) fx-4<3-2x
Solution
(a) x+.8>15
x>15-8
7
(b) 7-2x <11
-w4¢1-7
—2x <4
x>-2
() 3x-32-24x
Collect liketerms
1
gx-* 2-243
a
agree
_5
ae:
os
(4) 3x-1<2x—5
Collect liketerms
8X¥—-1<42%—5
3x—-2x<-5+1
x<-4
2x+8
(e) ze 20
Multiply by —3 both sides
2x +8 <—60
2x <—60-—8
2x = —68
we -34
99
() tx-4<3-2x
2 2
Collect liketerms
ae
gx-4S3-5x
: + ! 344
5xtRx s
Bxt4x og
a
7223
ars
=3(5)
eee 7,
_18
me
Linear Inequalities from Practical Situations
Form linear inequalities from practical situations
To represent an inequality on a number line, the following are important to be considered:
° The endpoint which is not included is marked with an empty circle
. ‘The endpoint which is included is marked with a solid circle
Example 12
(a) x>3
+ on | roe Te
—1 0 £2 ¥Y 4 5
(b) -1<x <3
cr ff foe Te
“1. 0 2 2 38 4
100
Compound statement —is a statement made up of two or more inequalities
Example 13
Solve the following compound inequalities and represent the answer on the number line
(a) 10<2x-3<14
(b) 7<53-—2x<15
Solution
(a) 10<2x-3<14
Express in the form ofa <x <b
Add 3 on each part of the inequality
10+3 <5 2x-34+3<144+3
13 2x<17
Divide by 2 each part of the inequality
13 cree 17
a “2
dgath
ee ee =4
ee
reap ge
“i @i28848 67 8 @
101
(b)7<$3-2x<15
Subtract 3 on each part of the inequality
VFS F-38522 13
4<-2x<12
Divide by —2 each part of the inequality
= 2x> ge
27 —2
-2>x>-6
—-6<x<s-2
—— rf & tf Lee
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
102