Equations in Science
3.1 Basic Techniques
3.1.1
Introduction
Equations in science frequently use letters to show how factors combine to produce a specific outcome.
As a simple example, we know that the area of a right-angled triangle is equal to half the base times the height. The formula giving the area can be written as an equation:
Area=1/2 × Base × Height
Essential Mathematics and Statistics for Science 2nd Edition Graham Currell and Antony Dowman
In algebra we often use single letters or abbreviations to ‘stand for’ the various factors in an equation, and hence we could write:
A= 0.5 × b × h
whereA= area, b = length of the base and h = height of the triangle.
In this section we introduce some basic algebraic techniques in the context of their use in handling simple equations.
Some students may find it useful to refresh their understanding of the basic mathematics associated with handling numbers, powers, fractions, etc., by visiting the Revision Mathematics section of the Website.
3.1.2
BODMAS (or BIDMAS)
The order for working out an algebraic expression follows the acronym BODMAS (or BID- MAS): Brackets first, then power Of (or Indices), followed by Divide or Multiply, and finally Add or Subtract.
Q3.1
Check that you understand the BODMAS rules, by replacing x with the num-
ber 2 and y with the number 3 in the following expressions, and calculate the
values: (i) 3x2 (v) 4(y− x2) (ii) (3x)2 (vi) 2xy2 (iii) 4(y− x) (vii) xy− yx (iv) x y2 (viii) x+ y x
3.1.3
Algebraic equations
We can illustrate the versatility of equations by considering the changing speed of a car travelling along a road. We can use just the one equation for several situations:
v= u + at [3.1]
wherev= final velocity, u = initial velocity, a = acceleration and t = time passed. (Equation
Example 3.1
In the first situation, the car is initially travelling at u= 10 m s−1 (approx. 20 mph), and then the driver accelerates at a rate ofa= 2 m s−2(increasing by 2 m s−1for each second) for a period of t= 10 seconds. The final velocity is given by:
Final velocity, v= 10 + 2 × 10 = 10 + 20 = 30 m s−1 (approx. 67 mph)
Example 3.2
In another situation, a high-performance car starts from rest (with an initial velocity
u= 0 m s−1) and then accelerates at a= 6 m s−2 fort = 4.5 seconds, giving
Final velocity = 0 + 6 × 4.5 = 27 m s−1 (0 to 60 mph in 4.5 seconds!)
We have used different values ofv, u, a and t to describe the different situations.
In this context,v, u, a and t are called the variables in the equation.
Q3.2
Use equation 3.1 to calculate the final velocity of a car which is initially going at 5 m s−1and then accelerates at a rate ofa = 3 m s−2for a period of 4 seconds.
Q3.3
Use equation [3.1] to calculate the final velocity of a car which is initially going at 30 m s−1and then accelerates at a rate of a= −5 m s−2(the negative sign shows that this is actually a deceleration) for a period of 6 seconds.
In some types of problems, one of the factors may have the same constant value for a range of different situations. For example, in Einstein’s well-known equation:
E= mc2 [3.2]
E is the amount of energy (in joules) released when a mass, m (in kg), of matter is completely
annihilated. E and m are both variables related by the constant c, which has a value of
Example 3.3
Use equation [3.2] to calculate the total energy produced when 1.0 mg (1 milligram = one-thousandth of a gram) is totally annihilated in a nuclear reactor.
Now 1.0 mg= 1.0 × 10−3 g= 1.0 × 10−6 kg, so
E= 1.0 × 10−6× (3.0 × 108)2= 9.0 × 1010= 90 000 MJ
(enough energy to power a 1 kW heater continuously for almost 3 years).
The separate parts of an algebraic expression are often called ‘terms’.
There are certain conventions which define how ‘terms’ in algebra should be written: • A term with a number times a letter, e.g. 5 × a, would usually be written just as 5a (the
number in a product goes before the letter, and it must not be written after the letter asa5).
• A term with ‘1×’ such as 1 × t would just be written as t without the ‘1’. • A product of letters, e.g. k × a, could be written as ka or ak.
• Sometimes a raised full stop is used to represent the multiplication sign between letters:
k× a (= ka) could also be written as k · a (but 3a would not be written as 3 · a).
Q3.4
Given that v= u + at, decide whether each of the following statements is either
true, T, or false, F:
Ift = 5, the equation could be written as v = u + a5 T/F
Ift = 5, the equation could be written as v = u + a × 5 T/F
Ift = 5, the equation could be written as v = u + 5a T/F
Ifa = 1, the equation could be written as v = u + t T/F
Very often in science, we use letters from the Greek alphabet (see Website) in addition to those from the English alphabet. For example:
v= f × λ
wherev is the velocity of a wave with a frequency, f , and a wavelength, λ (lambda), and:
in a time,t.
3.1.4
Negative values
The ‘sign’ of a term adds a ‘sense of direction’ to the value that follows it.
If, in a particular problem, we consider upwards to be the positive direction, then an object going upwards with a speed of 5 m s−1would have a velocity of+5 m s−1, whereas an object falling downwards with the same speed would have a velocity of−5 m s−1.
For a stone thrown vertically into the air, we can still use equation [3.1] withu= initial upward velocity andv= final upward velocity. In this case, the acceleration will be a = −9.8
m s−2, which is the constant acceleration due to gravity. It is written as negative here because we have chosen upwards as being the positive direction in this problem, but gravity acts
downwards.
The equation describing the upward velocities becomes:
v= u − 9.8t [3.3]
Example 3.4
If a stone is thrown upwards at 10 m s−1, how fast will it be travelling upwards after (i) t= 0.5 seconds?
(ii) t= 1 second?
(iii) t= 2 seconds?
Using equation [3.3]:
(i) v= 10 − 9.8 × 0.5 = 5.1 m s−1 the stone is slowing down
(ii) v= 10 − 9.8 × 1 = 0.2 m s−1 it has almost stopped
(iii) v= 10 − 9.8 × 2 = −9.6 m s−1 it is now falling downwards with
increasing speed
There are three main rules when using signs with algebraic terms: • The sign ‘goes with’ the term that follows it.
• If there is no sign, then it is assumed that the sign is positive.
• The order of the terms is not important as long as the ‘sign’ stays with its associated term.
Multiplying with negative terms follows the familiar rules: • a minus times a minus gives a plus;
• a plus times a minus gives a minus.
For example, in calculating the termsa× t and −a × t:
If a = −3 and t = −2 then a × t = (−3) × (−2) = +6
If a = −3 and t = −2 then −a × t = −(−3) × (−2) = −6
If a = 3 and t = −2 then a× t = 3 × (−2) = −6
If a = −3 and t = 2 then −a × t = −(−3) × 2 = +6
3.1.5
Brackets
Brackets (also called parentheses) are used to group together the terms that are multiplied by the same factor (in the example below the factor isa). For example:
2a+ 3a ⇒ (2 + 3)a ⇒ 5a
Similarly, counting ‘a’s in the expression xa+ ya gives us the equation:
xa+ ya ⇒ (x + y)a
i.e.x lots of ‘a’ plus y lots of ‘a’ give a total of (x+ y) lots of ‘a’.
We can also change the order of multiplication:
(x+ y)a ⇒ a(x + y)
When multiplying out brackets the factor outside the bracket multiplies every term inside the bracket:
a(x+ y) ⇒ ax + ay ⇒ xa + ya
(which equals the original expression above).
If the multiplying factor is a minus sign, then every term inside the bracket is multiplied by ‘−1’, and changes sign:
Example 3.5
Multiply out the brackets in the following expressions:
3(2x+ 3) ⇒ 6x + 9
v(v+ w) ⇒ v2+ vw
v(2x+ 3) ⇒ 2vx + 3v
−3(a − b) ⇒ −3a + (−3) × (−b) ⇒ −3a + 3b = 3b − 3a
−(4p − 3q) ⇒ (−1) × (4p − 3q) ⇒ −4p − (−1) × 3q ⇒ −4p + 3q ⇒ 3q − 4p
−(x − 2b) ⇒ −x + (−1) × (−2b) ⇒ −x + 2b ⇒ 2b − x
Q3.5
Multiply out the following brackets:
(i) 3(2+ x) (iv) p(x+ 2)
(ii) 3(2+ 4x) (v) −3p(2 − x)
(iii) −2(4x − 3) (vi) p(x+ p)
When multiplying out the product of two brackets, first multiply the second bracket sepa-
rately by every term in the first bracket, and then multiply out all the different terms as in
Example 3.6.
Example 3.6
To multiply out (v− t + d)(p − t), multiply the second bracket by every term in the
first bracket:
(v− t + d)(p − t) = v(p − t) − t(p − t) + d(p − t)
and then separately multiply out each of the three new terms in brackets: = vp − vt − tp + t2+ dp − dt
Q3.6
Multiply out the following brackets:
(ii) (v− t)(v + t) (v) (x− 2y)(p − q)
(iii) (v+ 4)(3 − vt) (vi) (v− t + 2)(3 + v + t)
Brackets are sometimes used in other ways. For example, the expression log(1+ x) is not
a multiplication of ‘log’ with(1+ x), but it is an instruction (3.3.5) to take the logarithm of (1+ x).
3.1.6
Reading equations
When ‘reading’ a scientific equation, it is important to remember that it is a mathematical
representation of relationships that may exist in a real-world system.
Example 3.7 shows how an equation can represent a general series of relationships, and how it is then possible to calculate a specific outcome, given a particular condition (i.e . the value ofn in this example).
Example 3.7
The statement ‘I think of a number,n, double it, add 6, and divide by 2 to obtain a final
number, N ’ can be represented by the equation: N =2n+ 6
2
If the original number,n, were 4, the answer, N , would be 7, and if the original number, n, were 6, the answer, N , would be 9.
Q3.7
Write equations forN which represent the following processes, and then use them
to calculate a result for the data supplied:
(i) I think of a number,n, and add 3, giving a value, N .
Use the equation to calculateN for n= 6.
(ii) I think of a number,n, double it, and then add 3, giving a value, N .
What isN if n= 2?
(iii) I think of a number,n, add 3, and then double the result to give a final value, N . What is N if n= 2?
(iv) I think of a number,n, add 3, double the result, take away 6, and then divide
by 2, to give a final value,N .
Example 3.8
The statement ‘For a given amount of an ideal gas, the pressure, p, is proportional to
the product of the absolute temperature, T , and the reciprocal of the volume, V ’ can be represented by the equation:
p∝ T × 1
V
‘Product’ implies multiplication in maths.
The ‘reciprocal’ of a value is ‘one over’ that value.
A ‘proportional’ relationship can be converted into an ‘equality’ equation by including a ‘constant of proportionality’; in this case we includek:
p= k × T × 1
V ⇒ k ×
T V
Note that forn moles of an ideal gas, k= n × R, where R is the gas constant.
Q3.8
Write equations to represent the following relationships.
For example, the statement ‘The area, A, of a circle can be calculated by multi-
plying the constant, π (Greek letter, pi), with the square of the radius, r’ can be
represented by the equation:A= πr2.
(i) The area,A, of a triangle is half the base, b, times the height, h.
(ii) The distance, d, covered by a runner is equal to the product of the average
speed,v, and the time, t.
(iii) The body mass index, BMI, of a person is equal to their weight (mass), m,
divided by the square of their height,h.
(iv) The volume,V , of a spherical cell is four-thirds the constant, π , multiplied by
the cube of the radius,r.
(v) The velocity,v, of an object that has fallen through a distance, h, is given by
the square root of twice the product ofh and g, where g is the acceleration
due to gravity.
3.1.7
Factors
Example 3.9
Check your understanding of each of the following:
‘3’ and ‘10’ are both factors of ‘30’: 30= 3 × 10
‘2’, ‘5’, ‘6’ and ‘15’ are other factors of ‘30’: 30= 2 × 15 and 30 = 5 × 6
‘2’ and ‘y’ are both factors of ‘2y’: 2y= 2 × y
‘2’, ‘5’ and ‘y’ are factors of ‘3y+ 7y’: 3y+ 7y = 10y = 2 × 5 × y
(2+ p) is a factor of ‘(2 + p)y’ (2+ p)y = (2 + p) × y
‘y’ is a factor of ‘(2+ p)y’ (2+ p)y = (2 + p) × y
‘p’ is not a factor of ‘(2+ p)y’ The expression cannot be rearranged to give p× (other factors)
Factorization is the process of splitting up an expression into its various factors.
Example 3.10
What are the factors of: 15xy+ 12yz?
‘3’ is a factor : 15xy+ 12yz = 3(5xy + 4yz)
‘y’ is also a factor : 3(5xy+ 4yz) = 3y(5x + 4z)
We can now also see that ‘(5x+ 4z)’ is another factor.
We check that the expressions are equivalent by multiplying out the brackets again, i.e. 3y(5x+ 4z) = 15xy + 12yz.
Example 3.11
What are the factors of: 2p2+ 8p?
‘2’ is a factor : 2p2+ 8p = 2(p2+ 4p)
‘p’ is also a factor : 2(p2+ 4p) = 2p(p + 4)
We can now also see that ‘(p+ 4)’ is another factor.
We check that the expressions are equivalent by multiplying out the brackets again, i.e. 2p(p+ 4) = 2p2+ 8p.
Q3.9
What are the factors in each of the following? (i) 4x+ 4y
(ii) 4x+ 6y
(iii) 4x+ 6x2 (iv) pqx+ pbx
3.1.8
Cancelling (simplifying)
It is possible to cancel a term from both the numerator (top) and the denominator (bottom) of a fraction – as long as that term is a factor of both the numerator and the denominator .
Example 3.12
Check your understanding of each of the following:
(i) 30
25 =
30 6
25 5 =
6
5 ‘5’ is a factor of both ‘30’ and
‘25’ and can be cancelled from top and bottom.
(ii) 8
4 =
82 41
=2
1 = 2 ‘4’ is a factor of both ‘8’ and ‘4’.‘2’ divided by ‘1’ is just written as ‘2’. (iii) 6x 2y = 63x 21y = 3x
y ‘2’ is a factor of both ‘6x’ and
‘2y’ and can be cancelled from top and bottom.
(iv) 3x 2x =
3x
2x =
3
2 ‘x’ is a factor of both ‘3x’ and
‘2x’ and can be cancelled from top and bottom.
(v) x(4+ p)
2x =
x (4+ p)
2x =
(4+ p)
2 Note that ‘4’ is not a factor of the
top and cannot be cancelled with the ‘2’ below.
(vi) (4x+ p)
2x cannot be simplified The ‘x’ cannot be cancelled
because it is not a factor of the numerator (top). (vii) 3(x+ p) x(x+ p) = 3(x+ p) x(x+ p) = 3
x The bracket (x+ p) can be
cancelled top and bottom – it is a factor of both.
Q3.10
Simplify the following equations by cancelling where possible: (i) x = 4a 2b (iii) x = y(4a+ 1) 2by (ii) x = 4ap 2pb (iv) x = 2y(2a+ 3) 4by
3.1.9
Dividing and multiplying fractions
In the same way that numbers or variables can be multiplied in any order, e.g.
a× b × c = b × c × a = c × a × b etc.,
the division of numbers or variables can also be performed in any order. In the following example a fraction is divided by a further variable:
a b c = a c b = a b× c etc.
For example, ifa= 12, b = 3 and c = 4: a b c = 12 3 4 = 4 4 = 1 and a c b = 12 4 3 = 3 3 = 1 and a b× c= 12 3× 4= 12 12 = 1
However, it is also useful to remember that dividing by a fraction is the same as multiplying
by the reciprocal of that fraction: c a b = c ×b a and 1 1 b = 1 × b 1 = b 1 = b
For example, ifa= 12, b = 3 and c = 4:
4 12 3 = 4 × 3 12 = 4× 3 12 = 12 12= 1 and 4 12 3 = 4 4 = 1
Q3.11
Decide whether the answer is Yes or No for each of the following questions, whatever the values ofx, y, a and b. None of x, y, a and b are zero.
(i) Does x× a a× y equal x y? (iv) Does x/a y equal x y× a? (ii) Does 2x 2y equal x y? (v) Does x y equal 1 y/x? (iii) Does x+ 2 y+ 2 equal x y? (vi) Does x/a y/b equal bx ay?