Unit 12 Section 2 : Substitution into Formulae 2

In this section we look at substituting positive and negative values into formulae, as well as more complex expressions involving powers and square roots. As part of this it is necessary to revise order of operations and operations on negative numbers.

Order of Operations

The two main methods taught for working out the order of operations are BODMAS and BIDMAS.

If more than one operation in a calculation has the same precedence, the operations are carried out from left-to-right.
For this chapter it is important to note that powers are calculated before multiplication. For example:
If a = 3 and b = 4, what is the value of abē ?
CORRECT ANSWER
 = abē
 = a Ũ bē
 = 3 Ũ 4ē
 = 3 Ũ 16 = 48		 WRONG ANSWER
 = abē
 = (a Ũ b)ē
 = (3 Ũ 4)ē
 = 12ē = 144
So in the case of abē we do the bē part before we multiply by a.

Operations on Negative Numbers

You can use a number line like the one below to help when adding and subtracting negative numbers:

The table below summarises what happens when we add or subtract negative numbers.

Adding and subtracting negative numbers
If you add a negative number you move to the left on a number line.
If you subtract a negative number you move to the right on a number line.
Examples of adding negative numbers
7 + (–4) = 3 (start at 7 and move 4 to the left)
1 + (–4) = –3 (start at 1 and move 4 to the left)
–3 + (–4) = –7 (start at –3 and move 4 to the left)
Examples of subtracting negative numbers
4 – (–3) = 7 (start at 4 and move 3 to the right)
–2 – (–3) = 1 (start at –2 and move 3 to the right)
–8 – (–3) = –5 (start at –8 and move 3 to the right)
Check the examples yourself on the number line above.

Below is a summary of what happens when you multiply or divide negative numbers.
We can summarise the rules for multiplying and dividing two numbers as follows:
If the signs are the same (both positive or both negative), the answer will be positive.
If the signs are different (one positive and one negative), the answer will be negative.
For this chapter you should note that squaring a negative number (multiplying it by itself) will always give a positive answer.
For example:
If a = (–3), what is the value of aē ?
CORRECT ANSWER
 = aē
 = a Ũ a
 = (–3) Ũ (–3) = +9		 WRONG ANSWER
 = aē
 = –3ē
 = –(3 Ũ 3) = –9
So in the case of abē we do the bē part before we multiply by a.

Square Roots

The square root of a number is the value which we would square (multiply by itself) to get that number.
For example, the square root of 25 is 5, because 5ē = 5 Ũ 5 = 25.

The square root has a special symbol – we write it like this: = 5

You should be able to find the square root button on your calculator – it looks like the symbol above.
If you have a square root in a formula, work out everything inside the square root before you do the square root.

For example:

 

Practice Questions

Work out the answer to each of these questions then click on the button marked Click on this button below to see the correct answer to see whether you are correct.

If a = 6, b = 5, c = -2 and d = -3, determine the value of:











 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
Calculate:
(a)6 + (–2)

(b)(–3) + 5

(c)(–4) + (–2)

(d)2 – 4

(e)3 – (–2)

(f)(–7) – (–4)

(g)2 Ũ (–6)

(h)(–10) Ũ 5

(i)(–12) Ũ (–4)

(j)(–8) ũ 4

(k)14 ũ (–7)

(l)(–25) ũ (–5)

(m)(–3)ē

(n)(–5)ē Ũ (–2)

(o)(4 Ũ 5) + (–2)

(p)(–3) Ũ (–4) ũ 6

(q)(–3) Ũ (–8) + (–7)

(r)
(–6) Ũ (–4)
(–12)

(s)
(–10)ē
4

(t)(–3) Ũ (–5) Ũ (–9)

(u)(–5)ē + (–6)ē

Question 2
If a = 6, b = 3, c = 7, calculate:
(a)ab
(b)b + c
(c)c – a
(d)4b + 6c
(e)4c – 2b
(f)6a – 2c
(g)abc
(h)ab – bc
(i)2bc + ac
(j)bē
(k)aē – bē
(l)aē + bē – cē
Question 3
If a = 2, b = –4 and c = –5, evaluate:
(a)aē + bē
(b)ab
(c)bc
(d)a – b
(e)c – b
(f)3a + 2c
(g)2a – 4c
(h)3a + 2b
(i)ab – ac
Question 4
Calculate the value of the expression below when a = 15, b = 2 and c = –3.

Question 5
A formula for the perimeter of a triangle is p = x + y + z, where x, y and z are the lengths of the three sides.
Calculate the value of p when x = 1.5 cm, y = 2.5 cm and z = 3.5 cm.

cm

Question 6
The area of a trapezium is given by the formula below:
Calculate the area of the trapezium for which a = 3 cm, b = 3.6 cm and h = 2.2 cm.

cmē

Question 7
The length of one side of a right-angled triangle is given by the following formula:
Calculate the length l, if h = 13 cm and x = 12 cm.

cm

Question 8
The following formula can be used to convert temperatures from degrees Celsius (C) to degrees Fahrenheit (F):
Calculate the value of F, if:
(a)C = 100
(b)C = 20
(c)C = –10
(d)C = –20
Question 9
A formula states that:
Calculate the value of s, if:
(a)u = 3, v = 6 and t = 10
(b)u = –2, v = 4 and t = 2
(c)u = –10, v = –6 and t = 3
(d)u = –20, v = –40 and t = 3

You have now completed Unit 12 Section 2
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Produced by A.J.Reynolds October 2007