Change of Subject of Formulae
A formula is an equation which specifies how a number of variables are related to one another. Formulas are written so that a single variable, the subject of the formula, is on the left hand side of the equation. Everything else goes on the right hand side of the equation.
The volume of a cylinder is given thus V = πr2h, in this example, V is expressed in terms of the r and h or we say that V is made the subject of the formula.
Sometimes it becomes necessary to change the subject of the formula. To do this we regard the formula as an equation and solve for the letter which is to be the subject. For example in the above formula, we can make h the subject as follows:
V/πr2 =πr2h/πr2 = h
\ h = V/πr2, this is the same as expressing h in terms of v and r.
When you do a question yourself it is often helpful to write in these key points before you do the actual algebra.
Example : Make x the subject of y=x−5
Add 5 to both sides
Example: Make x the subject of y=8x
Divide both sides by 8
8 x 8=y8
x = y8
Make x the subject of y=xm+c
xm + c=y
Fractions are more difficult to work with, so to make work easier and errors less, likely get rid of the fraction first.
Multiply EVERYTHING on both sides by m
m(xm) + m(c)= m(y)
X + cm=my
Subtract cm from both sides
x=my − cm
- Make y the subject of the formula x = (y + z)/(y – z)
x = (y + z)/(y – z)
x (y – z) = y + z[multiply both sides by (y – z)] xy – xz = y + z
xy – y = z + zx
y (x – 1) = z (x + 1)
y = z(x + 1)/(x – 1)
In everyday life, we come across relationship between quantities. These relationships can be considered in varying one quantity and knowing how the other quantities vary. For example, the variation in the cost of things and their quantities, or the variation in the distance travelled by car and the time taken, or the variation in the amount of money a school receives and the population of a school, to mention but a few.
These variation of quantities can be in many forms depending on the quantities concerned.
Basically there are four forms of variations, namely:
Direct, Indirect (inverse), Joint and Partial Variations
Now consider two quantities x and y, if when y increases x increases also; or when y decreases x decreases in the same proportion, then we say the quantity y and x are in direct variation, or that y varies directly as x, or that y is directly proportional to x, this is written as y α x, where α is the sign of variation or proportionality.
Examples of Direct Variations
- Consider the quantities:
N = Number of tins of milk and C = Cost of the tins of milk, suppose that 1 tin of milk costs ₦40, then 2 tins will cost ₦80, 3 tins will cost ₦120, 4 tins will cost ₦160, 5 tins will cost ₦200, etc.
Tabulating this we have:
Tins of milk (₦) 1 2 3 4 5 Cost (C) 40 80 120 160 200 N/C 1/40 1/40 1/40 1/40 1/40
We can see that the number of tins increase, the cost also increases. So the number of tins of milk N and C vary, they can be called variables. Thus we say that the variable N is directly proportional to the variable C. Notice from the table that in the case N/C = 1/40 is a constant called the constant of variation or proportionality.
Thus we have N = C/40 (i.e. C x 1/40).
This is the equation relating or connecting N and C which can be used to determine the cost of any given number of tins or the number of tins a given amount can buy. This relation is sometimes called the law of variation
Suppose that a car travels 10km in 10 mins, then with the same speed it will travel: 20kn in 10 min, 30 km in 15 mins, 40 km in 20 mins, 50 km in 25 mins, etc.
Tabulating this we have
Distance d (Km) 10 20 30 40 50 Time t (mins) 5 10 15 20 25 d/l 2 2 2 2 2
We would observe that as the distance d increases time t also increases, or the distance decreases time t also decreases, and so the distance d varies directly as the time t.
i.e. d α t
Also d/t = 2 is the constant of proportionality which can be used to find out how far the car travels at any given time or the time it will take the car to cover the given distance.
Note that distance/time = speed
\ d/t = 2 is the speed of the car which is constant.
Note also that if the speed is not constant, then the distance will not be directly proportional to time as can be seen in the next example.
Example: The table below indicates the distance travelled by a car at various times.
Distance d (Km) 10 20 30 40 50 Time t (min) 6 10 20 25 30
Determine if the distance is directly proportional to time.
Distance d (km) 10 20 30 40 50 Time t (min) 6 10 20 25 30 d/t 5/3 2 3/2 8/5 5/3
If d is directly proportional to t, i.e. d α t, then d/t must be a constant. We can see from the table above that the values d/t are different and therefore d is not directly proportional to t. So we conclude that two quantities x and y are in direct are in direct variation if one quantity x varies in the same ration (proportion) as the other quantity y. i.e. x and y are in direct variation if x/y or y/x is a constant.
If when one quantity y increases, the other quantity x decreases and vice-versa in the same ratio then we say that the two quantities x and y are indirectly or inversely or that y varies inversely as x or that y is inversely proportional to x written.
For example the time taken to do a job is inversely proportional to the number of men that will do the job since as the number of men increases the time to do the job will decrease.
The table below shows the time taken to travel from Abuja to Lagos at different speeds.
Time t (hrs) 20 10 8 5 4 Speed s (Km/h) 20 40 50 80 100 Distance s x t 400 400 400 400 400
It can be seen from the table above that as the speed increases the time taken decreases, as the speed reduces the time increases, also in each case s x t = 400 which is constant. Hence we say that the speed varies inversely with time. Since s x t = 400, means s = 400/t. This is the equation of the relationship between s and t. So in general, if y varies inversely as x, i.e. y α 1/x then y = k/x, where k is the constant of proportionality.
In this variation, three or more quantities or variables are involved in the relationship which occur in many forms. These forms might involve the combination of two direct variations or the combination of one direct and one indirect, etc., as can be seen in the following examples.
Examples of Joint Variation
- z α x and z α y are two direct variations which can be combined to form a joint variation z α xy which is read as z varies jointly as x z and y.
\ z = kxy where k is the constant of variation.
- F α Ml/d2 Þ F = kMl/d2 where k is the constant – a joint variation involving two direct and one inverse variations F α M, F α l and F α 1/d2.
Partial or part variation consists of two or more parts of quantities added together, one part may be constant while the others can vary either directly, indirectly or jointly.
Examples of partial variation
- S is partly constant and partly varies as T is written as S α k +T, where k is the partial constant.
\ S = k + cT where k and C are both constants.
- E partly varies directly as M and partly varies jointly as M and v is denoted as E α M + Mv2
This implies E = kM + cMv2 where k and c are constants.
In partial variation at least two constants such as k and c above are involved. These constants are to be found when finding the equation relating the variables.
Lets see how much you’ve learnt, attach the following answers to the comment below
The table below indicates the time taken for a train to pass through tunnels of different lengths.
Length l (Km) 1 2 3 4 ½ Time t (sec) 30 60 90 135
- What is the constant of proportionality? A. 1/40 B. 2/40 C. 1/30 D. 2/30
- How far will the train go in 45 secs? A. 1.4 B. 1.5 C. 1.6 D. 1.7
y is inversely proportional to x. If y = 2.5 when x = 5
- What is the value of y when x is 6.25? A.3 B. 2 C. 4 D. 5
- What is the value of x when y = 50? A.0.25 B. 0.26 C. 0.24 D.0 5
- In the relation C/5 = make (F – 32)/9 make F as the subject. A. 9C/5 + 32 B. 9C/5 – 32 C. 5C/9 + 32 D. 5C/9 – 32