Chapter 4 Algebraic Formulae

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CHAPTER 4 : ALGEBRAIC FORMULAE

1. A variable is a quantity where its value is not fixed.

2. A constant is a quantity where its value is fixed.

3. An algebraic formula is an equation that relates several variables with

constants.

4. The subject of an algebraic formula is a variable expressed as a term in terms

of other variables.

5. An algebraic formula can be rearranged to let another variable to be the

subject of the formula. The process involves:

(a) change the operations: addition, subtraction, multiplication and division;

(b) change roots and square roots

(c) combined operations and determination of roots and power.

A. VARIABLES AND CONSTANTS.

State whether each of the following quantities is a variable or a constant.

Example:

The number of sides of a pentagon.

Constant.

Exercise:

1) The age of your parents.

2) The number of colours of the

rainbow.

3) The daily sales of PTA Stationaries.

Algebraic Formulae 42

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State the possible values for each of the following variables.

Example:

The mass (m) of a watermelon, in

kg.

2.5 kg

Exercise:

4) The volume of food (V) consume by

your friends, everyday, in g.

5) The number of students who

scored A’s in Mathematics in

the PMR.

6) The water charges (q) per month of a

school, in RM.

B. FORMULAE

Write formulae based on statements and situations.

An iron pipe measures 95 cm long while a PVC pipe is 7.5 m long. Write a formula for the total length L, in m, of b iron pipes and p PVC pipes.

Errors

L = 95b + 7.5p

Correct Steps

L = 0.95b + 7.5p


Unit in m

Incorrect because unit is in cm

Units are in m

Correct because unit is in m

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Example:

The entrance fees to a museum are RM4 for an adult and RM3 for a child. If, on a certain day, p adults and q children visited the museum, write a formula for the total collection of that day.Solution:

Let A = Total collection (in RM)

A = (Number of adults × 4) +

(Number of children × 3)

A = 4p + 3q

Exercise:

1) Saiful has RMp. After spending RM(q + 2) per day for a week, the remainder is RM18. Find a formula for p.

2) Syafik is b years old. His mother is twice his sister’s age. If Syafik is 5 years younger than his sister, write a formula for the sum (S) of their ages.

3) The price of a bicycle is RM500. Write down a formula for the new price, N, if the price is increased by r%.


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Example:

Write a formula relating m and n based on the table below.

M 1 2 3 4 5

N 3 5 7 9 11

Solution:

From the table:m = 1 => n = 2(1) + 1 = 3m = 2 => n = 2(2) + 1 = 3m = 3 => n = 2(3) + 1 = 3׃ ׃ ׃ ׃ ׃ ׃ Hence; n = 2m + 1

4)

Write a formula relating m and n based on the table below.

m 1 2 3 4 5

n 3 6 9 12 15

Solution:

5)Write a formula relating m and n based on the table below.

m -1 0 1 2 3

n -3 -2 -1 0 1

Solution:

6)Write a formula relating m and n based on the table below.

M 0 1 2 3 4

n -5 -3 -1 1 3

Solution:


Values of m are substituted here

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C. Change the subject

1) Given that 3m – = 1, express m in terms of n.

Common Errors

Error

3m – = 1

3m – 2n = 3

3m = 3 + 2n

m =

Correct Steps

3m – = 1

9m – 2n = 3

9m = 3 + 2n

m =

2) Given , express u in terms of v

Common ErrorErrors

u + v = 5

u = 5 – v

Correct Steps

5v + 5u = uv

uv – 5u = 5v

u(v – 5) = 5v

u =

3) Given that a2 – b2 = 9c2, express a in terms of b and c.


Incorrect because 3m is not multiplied by 3

Incorrect because the concept of LCM is not applied

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Common ErrorError

a2 – b2 = 9c2

a2 = 9c2+ b2

a = 3c + b

Correct Steps

a2 – b2 = 9c2

a2 = 9c2+ b2

a =

4) Given that 2 = 3m, express h in terms of k and m.

Error

2 = 3m

2(h + k) = (3m)2

(h + k) =

h =

Correct Steps

2 = 3m

[2 ]2 = [3m]2

4(h + k) = 9m2

(h + k) =

h =

Exercise:

1) Express a in terms of w, if

w =

2) Given a = , express b in terms of a

and c


Incorrect because√(9c2+ b2) ≠ √9c2 + √ b2

Incorrect because2 should be squared

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3) Given p = , express w in

terms of p and v

4) Given 2k – = m, express m in terms

of k and n.

5) Given 3(x – z) = 4xy, express x

in terms of y and z.6) Given , express u in terms of f

and v.

7) Given 5pq = qr – 2p, express p

in terms of q and r.8) Given 2π = T, express g in terms of

T and

9) Given v = , express x in

terms of v.

10) Given 5 = t, express n in terms

of m and t.


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D Finding the value of a variable.

Example:

Given that W = 3a – , find

(a) W when a = 2 and b = 4,(b) b when a = 3 and W = 5.Solution:

W = 3a –

(a) Substitute a = 2 and b = 4 into

the formula

W = 3(2) –

= 6 – 2

= 4

(b) 5 = 3(3) –

= 9 – 5

b = 4 × 2

= 8

Exercise:

1) Given p = 3q + 5r, find the values of

(a) p when q = 4 and r = 3(b) q when p = 7 and r = 1

Solution:

2) Given k = 2m – 3n2, find the

values of

(a) k when m = 3 and n = -2(b) m when k = -5 and n = 3

Solution:

3) Given y = 2r – 3s + 6t, find the values

of

(a) y when r = 3, s = -2 and t =

(b) t when y = 4, s = 3 and r = -1Solution:

PMR past year questions

2003

1). Given that 3h = 5k + 4, express k

2004


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in terms of h.

[2 marks]

2). Given that , express p in

terms of k.

[2 marks]

2005

3). Given , express y in terms

of p.

[2 marks]

2006

4). Given , express n in terms of

F and k.

[2 marks]

2007

5). Given that r2 + 7 = p, express r in

terms of p.

2008

6). Given qr + s = p, express r in terms of

p , q and s.


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[2 marks] [2 marks]

CHAPTER 4 : ALGEBRAIC FORMULAEANSWERS

A. Variable and Constant1) Variable 2) Constant

3) Variable 4)

5) 20 students 6) RM300

B. Formulae1) p = 7(q + 2) + 18 2) S = 4b + 15

3) N = 500 + 5r 4) n = 3m

5) n = m – 2 6) n = 2m – 5

C. Change The Subject1) 2)


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3) 4)

5) 6)

7) 8)

9) 10)

D Finding the value of a variable

1)a) P= 27b) q = 2/3

2)a) k=-6b) m=11

3) Y=15

T =

PMR past year questions

20031). 5k = 3h - 4

k =

1

1

20042). 2p - 6 = 5k 2p = 5k + 6

p =

11

1

20053). 4y - 3 = 2y + 2p 2y = 2p + 3

y =

11

1


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20064). 5F = 3n - nk n(3 - k) = 5F

n =

11

1

20075). = p - 7

r =

11

20086). qr = p - s

r =

1

1


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Chapter 4 Algebraic Formulae