TECHNICAL MATHEMATICS 1

UNIT 1 : EQUATIONS AND FORMULAE
1.1 ALGEBRAIC EQUATIONS

Equation are mathematical statements
with the equal sign ( = ).

LEARNING OUTCOMES Example : Operation/sign

After completing the unit, students 2x + 1 = 5
should be able to:
coefficient
1. Solve algebraic equations with
one unknown. unknowns number/constant

2. Solve simultaneous equations Algebraic equations are equations that
with two linear equations. involve numbers and unknowns.

3. Transpose a formula to .
change its subject. Algebraic equations which its unknown
with one degree also known as linear
4. Evaluate a given formula. equation.

Solutions of algebraic
equations are finding the value
of the unknown in the equation.

Tips to remember : Unit 1: Equation And Formulae

 A positive term becomes a negative term when it
changes side of the equation.

 A negative term becomes a positive term when it
changes side of the equation.

 A term that is multiplied becomes a term that is
divided when it changes side of the equation.

 A term that is divided becomes a term that is
multiplied when it changes side of the equation.

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UNIT 1 : EQUATIONS AND FORMULAE
Example 1.1
Solve for x in the following equations :
a) x + 5 = 10
b) 2x – 1 = 21
c) 5y - 12 = 3y + 4

Solution : x+5 = 10 Subtract 5 on both sides of equation,
a) x+5 –5 = 10 – 5 and only unknown x is left on the LHS.
= 5
x Add 1 on both sides.

b) 2x – 1 + 1 = 21 + 1 Divide by 2 on both sides.
2x = 22

2x = 22
2 2
11
x=

c) 5y – 12 – 3y = 3y + 4 – 3y

2y -12 = 4 SuAbdtrdac1t23oynobnobthotshidseidses
2y – 12 + 12 = 4 + 12

2y = 16

2y = 16 Divide 2 on both sides. Unit 1: Equation And Formulae
22

y=8

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UNIT 1 : EQUATIONS AND FORMULAE 2) 2x - 3 = 11
Practice 1.1 :
Solution of Algebraic Equations.
Solve each of the following.

1) x + 5 = 9

3) 4x - 2 = 18 4) 80 - 3x = 7x

5) 5x = 30 6) k = 8 Unit 1: Equation And Formulae
7) − d = 14 5

7 8) 3x = 18
9) 7x + 20 = 6 10) 6x – 17 = -35
11) 9m + 5 = 5m - 7 12) 11 – 3n = 2n - 9
13) 5h + 30 = 7h 14) 3x – 24 = 9x

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UNIT 1 : EQUATIONS AND FORMULAE 16) 5x + 6 = 3x - 8
15) 7x - 5 = 3x + 15 18) 14 – 3n = 2n+ 9

17) 5m -12 =3m + 4

19) 5h - 36 = - 7h 20) 80 + 3x = 7x

21) 9x + 2 = 3x + 14 22) 5+4x-7 = 2x + 8
23) 5x - 6 = - 3x - 8 24) 5(h + 6) = 7h
25) 6 − 4( + 3) = 2( − 1)
27) 5(x + 2) = 3x - 8 26) 7 + 4 = 11
3

28) 3 (12 − 8 ) = −21 Unit 1: Equation And Formulae
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UNIT 1 : EQUATIONS AND FORMULAE
1.2 SIMULTANEOUS EQUATIONS

Sometimes we need to find the solutions of a pair of equations. There are many problems can
be solved by using two equations in two unknowns and solving them simultaneously. There are
many methods to solve these equations but in this unit we only focus to two methods, namely:

i.Elimination Method
ii.Substitution Method

ELIMINATION METHOD

Make the coefficients for one of the variables numerically equal.

Eliminate one variable by addition or subtraction

Solve the equation for the value of one variable, then the other.

Example 1.2 :

Use the elimination method to solve the following pairs of simultaneous equations.

3m – 4n = 10

3m – 2n = 8

Solution:

3m – 4n = 10 ------- (1)
3m – 2n = 8 ------- (2)

(1) – (2) : -2n = 2

n= 2
−2

= -1 IMPORTANT NOTES :

Substitute n = -1 into equation (2) :  make sure the coefficients for one Unit 1: Equation And Formulae
variable are equal.
3m – 2(-1) = 8
 subtract the two equation if both
3m + 2 = 8 variables have the same symbols.
3m = 8–2
 add the two equations if both
3m = 6 variables have different symbols.

m= 6
3

=2

Hence, the solution is m = 2 and n = -1.

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UNIT 1 : EQUATIONS AND FORMULAE
SUBTITUTION METHOD

Express one variable in term of the other in one equation.

Substitute the equation into the other equation.

Solve the equation for the value of one variable, then the other.

Example 1.3 :

Use substitution method to solve the following simultaneous equation.

x + 2y = 13
x – 3y = 3

Solution :

x + 2y = 13 -------- (1)
2x – 3y = 3 -------- (2)

From equation (2) :

x= 3 + 3y -------- (3) Express x in term of y.

Substitute x = 3+3y from equation (3) into equation (1) :

3+3y +2y = 13

5y = 10 y=2

From equation (3) :

x = 3+ 3(2)

x = 3+6

x =9 Unit 1: Equation And Formulae

Hence, the solution is x = 9 and y = 2.

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UNIT 1 : EQUATIONS AND FORMULAE
Practice 1.2 :

A. Solve the following simultaneous equations by the elimination method.

1) x + 3y = 15 2) a + 6b = 5
x +y = 7 a + 3b = 2

3) 2p + q = 1 4) 3x – y = 6
p+q=2 x–y=8

5) 3p - 2q = 10 6) 4x - y = 5
p - 2q = 2 3x - y = 2

7) 10h - k = 7 8) x + 2y = 5 Unit 1: Equation And Formulae
6h + k = 9 x – 3y = 15

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UNIT 1 : EQUATIONS AND FORMULAE 10) 5y + x = 21
2y + x = 3
9) 3x + y = 7
y+x=9

B. Solve the following simultaneous equations by the substitution method.

1) x + 2y = 5 2) a + 3b = 1
x – 3y = 15 a+b=3

3) x + 3y = -8 4) 2p + q = 1 Unit 1: Equation And Formulae
3x – y = 6 p+q=2

5) 2x + y = 3 6) 3h - k = 7
3x + y = 4 5h + k = 9

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UNIT 1 : EQUATIONS AND FORMULAE

1.3 FORMULAE

A formula is an equation that shows relationship between a few variables.

The subject of a formula is the variable that is expressed in terms of other variables. It
always appears on the left hand side of the equal sign, “=”.

For example, for the formula K = 1 mv 2 , K is the subject and K is expressed in terms of m
2

and v. K is the subject of formula

Example 1.4 :

a) If F = ma, express m in terms of F and a.

b) Make I the subject of the formula V = E −Ir
c) Given p = F , make F the subject of formula

A
Solution:

a) F = ma Interchange the sides
ma = F

a= F
m

b) V = E – Ir Add to both sides
V +Ir = E −Ir +Ir Subtract from both sides
V +Ir = E Divide both sides by

V + Ir – V = E – V Unit 1: Equation And Formulae

Ir = E − V
rr
I= E−V

r

c) p = F Interchange the sides
A Multiply both sides by

F =p
A
F=pA

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UNIT 1 : EQUATIONS AND FORMULAE

Example 1.5

Given the formula V = IR ,

a) Make I the subject of the formula
b) Find the value of I if V = 120, and R = 200.

Solution :
a) Solve for

V =IR
I =V

R

b) Substitute the data
I =V
R
= 120
200

I = 0.6

Example 1.6

A drill draws a current, , and the resistance, R. The power, P, in watts is given as P = I2R

a) Make R as the subject of the formula.
b) Given the current is 4.5 A and the power is 324 watts. Find the value of

resistance, R.
Solution :

a) P = I 2R IMPORTANT NOTES :
R= P
I2  Change or transpose the subject
of the formula
b) R= P Unit 1: Equation And Formulae
I2  Evaluate the formula.

R = 324
4.52

R = 16

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UNIT 1 : EQUATIONS AND FORMULAE

Practice 1.3 :

A. Change the subject for each of the following formulae to the variable stated in the
brackets.

1) g = 3 + h; [h] 2) V = π 2ℎ ; [h] 3) a = my2; [m]

4) 6p = 7m + 1; [m] 5) ax + y = z ; [x] 6) E = mgh ; [g]

7) m = 3n – 4k ; [n] 8) e = t ; [k] 9) m = 3(2h + k); [h]
5k

10) 7 = 2; [k] 11) 3p=4q+5 ; [q] 12) 5p − 4m = 2; [p]
3k+h 3

Unit 1: Equation And Formulae

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UNIT 1 : EQUATIONS AND FORMULAE

B. Answer the following question.

1) Given that = − , express x in term of y.


2) Make k the subject of the formula = + .


3) Given that 3 − 4 = , express n in terms of m.

4) Change the subject formula ℎ + 5 = 2ℎ, so that h becomes the subjects.
3

5) Given that the perimeter of a rectangle is P cm and the length is l cm. If the area of the
rectangle is 64cm2, express P in terms of l.

6) Given that S = a + (n-1)d, find the value of S if a=- 4, n=12 and d = 3.

7) Given that = + . Find the value of w if r = 4, and s = 3.

8) Given that = − + . Find the value of y if p = 5, q = -1 and r = 3. Unit 1: Equation And Formulae

9) Given = √ − , find y when x = 4 and z = 2.


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UNIT 1 : EQUATIONS AND FORMULAE Unit 1: Equation And Formulae
10) The surface area of cube is given by the formula S = 6 2.

a) Change as the subject of formula.
b) Hence, calculate the value of if given S = 54cm2.

11) The volume of a cylinder is given by the formula V = πr 2h , where is
the radius, and is the height. Find the height if the volume is
8550 m3 and the radius is 15.0 m.

v2
12) P watts of an electrical circuit is given by the formula P = .

R
Find the power when V is 14.8 volts and R is 19.5 ohms.

13) Distance is given by s = ut + 1 at2 . Given that s = 500, t = 4, a = - 5,
2

find u.

14) The surface area of pyramid is given by the formula S = 2 h + 2.
a) Express h in terms of S and .
b) Hence, calculate the value of h if given S = 32cm2 and = 4 cm.

15) The area of sector is given by the formula A = 1 r 2 .
2

a) Express  in terms of A and r.
b) Hence, calculate the value of  if given A = 60 cm2 and r = 10 cm.

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