The use of symbols

Open Sentences

Consider the question:

“What number added to 7 will give you 12?” How will you write that as a Mathematical sentence?

A Mathematical sentence consists of numbers and symbols (that represent unknown numbers) called variables. The equivalent Mathematical sentence of the above question can be written as:
+ 7 = 12

We call the above equation an open sentence.

Example:

14 +        = 17

This will be true if 3 goes in the box.

14 +   3     = 17

We say 14 +         = 17 is an open sentence. Any value can go in the box but usually only one value will make an open sentence true.

CLASS ACTIVITY

In each sentence, find the number which makes it true.

(i) 3 + 2 =

(ii) 20 − 14 =

(iii)        = 14 −

(iv)         ×          = 1

(v)         ×         = 49

Letters to Represent Symbols or the Unknown

In mathematics, we use letters of the alphabet to stand for numbers instead of boxes.

We write 14 + x instead of 14 +

Any letter can be used but capital letters are not used.

Examples:

Find the number which will make the following expression true.

(a) x + 2 = 7

(b) 2 x a – 5 = 3

(c) b2 = −2

Solution:

(a) The expression x + 2 = 7 will only be true if x is 5 i.e. 5 + 2 = 7.

(b) The expression 2 x a − 5 = 3 will only be true if a is 4 i.e. 2 x 4 – 5 = 3.

(c) The expression b2 = −2 will only be true if b is −4 i.e. −42 = −2

CLASS ACTIVITY

If each sentence is true, find the number which makes it true.

(i) x = 2 + 9

(ii) y = 9 + 5

(iii) q × q = 4

(iv) h × 4 = 20

(v) z – 6 = 2

(vi) g + g = 30

Open Sentences with Two Unknowns

Example:

If x = 5 and y = 3, find the value of:

(a) x + y

(b) x – 3 × y

Solution

(a) x + y

5 + 3 = 8

(b) x – 3 × y

5 – 3 × 3 = 5 – 9 = –4

CLASS ACTIVITY

If each sentence is true, find the value of a = 15 and b = 6

(a) b6 + a

(b) a × b

(c) a + b

(d) a – 3 × b

ASSIGNMENT

(a) k – 9 = 0

(b) 5 × x = – 50

(c) m + 4 = 11

(d) 3 × x+ 5 = -1

(e) z3 = 5

(f) 3 × x + 5 = -1

PRACTICE QUESTIONS

1. If y = 2 × x+ 3, find the value of y when:

(a) x = 5  (b) x = 9  (c) x = -5

2. If 2p + q = 7, find the value of p if:

(a) q = 1  (b) q = 3  (c) q = 5

3. (a) There are 21 matches left in a matchbox. Originally there were mmatches in the box and 27 have been used. What is the value of m?
4. The cost of a new water pump for a school community is ₦250 000. There are npeople in the community. When the cost is shared equally among the community members, they each have to pay ₦250 What is the value of n?
5. Find the value of the following when z = 11

(a) z + z  (b) z – z  (c) 9 + (z ÷ z)  (d) z + (5 × z)