GCSE: Changing the Subject

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified: 30th August 2015

The formula to calculate a temperature in Fahrenheit if we have the temperature in Celsius:

How could we find a new formula that allows us to determine the temperature in Celsius given the temperature in Fahrenheit?

?

Motivation

Basic Skill #1: ‘Undoing’ to UnlockMake the subject of the formula. Undo the last thing done to the subject each time

by doing the opposite.

𝑦=𝑥−2 𝑥=𝑦+2

𝑦=3 𝑥+2 𝑥=𝑦−23

𝑦=√𝑥+1 𝑥=(𝑦−1 )2

𝑦= 𝑥2−𝑎4

𝑥=±√4 𝑦+𝑎

?

?

?

?

Bro Tip: It doesn’t matter what side the subject is on, provided it’s on its own!

Basic Skill #2: Subject trapped in a negative term

When the subject is within the first argument of a subtraction, it’s easy to ‘release’.

𝑦=2 𝒙−3 2 𝒙=𝑦 +3?

However, it’s a tiny bit harder if the subject is in the term being subtracted.

𝑦=3−2𝑥 𝑦+2𝑥=32 𝑥=3− 𝑦

When the subject is inside a negative term, just add it to both sides.

?

?

Doing it in one step… (if you like)

How could you rearrange the numbers in to get another subtraction?

This suggests you can swap the thing you’re subtracting with the result. (i.e. Only the thing to the left of the subtraction stays put)

?

Examples:

𝑎− 𝑥=𝑏??

?

Exercise 1In each case make the subject of the formula. (Set 1 & 2: Only do odd questions)

1

2

3

4

5

6

78910

15

16

17

N

N

18

?

??

?

?

?11

12

13

14

????

??????

????

𝑥=𝑞𝑦−1

Basic Skill #3: Subject trapped in a denominatorWhen the subject is in the numerator of a fraction, it’s easy to ‘release’ the subject from the fraction.

𝑦=𝑥𝑞 𝑥=𝑞𝑦?

But it’s a bit harder if the subject is in the denominator…

𝑦=𝑞𝑥+1 𝑦 (𝑥+1 )=𝑞

𝑥+1=𝑞𝑦

In general, whenever you have a fraction in an equation, your instinct should be to multiply both sides by the denominator.

?

?

?

Basic Skill #3: Subject trapped in a denominator

! Isolate the fraction on one side of the equation, then multiply by denominator.

𝑎=𝑏−𝑐𝑥

𝑎𝑥

+𝑏=𝑐

𝑎=𝑏−𝑐𝑥

1 2

3

??

?

+2 first as was last thing done to

Doing it in one step… (if you like)

How would you rearrange the numbers in to get another division?

?

Examples:

𝑐=𝑏

4−2 𝑥4−2 𝑥=

𝑏𝑐

𝑎=𝑏𝑥

𝑎−𝑏2

𝑥+1=𝑐 𝑦= 𝑦 2

𝑥+𝑦−2

E1 E2E3

? ? ?

?

Thus we can swap the thing we’re dividing by and the result. The numerator is left unchanged.

Skill #3b: ‘Cross multiplying’

If you have just a fraction on each side of the equation, you can ‘cross multiply’.

𝑎𝑏

𝑐𝑑¿ Click for

Bromanimation

Examples:Make the subject:

? ?

E1E2

Exercise 2In each case make the subject of the formula. (Sets 1 & 2: Odd numbered questions)

1

2

3

4

5

6

7

8

12

13

14

N

?

??

?

?

?

?

?

?

?

?

?

?

9

11

N

?

?

(4 marks)

What does it mean to make the “subject of the formula”?

Can we think of generic tips that will help us solve these kinds of questions?

A* Question

Tip Why? When?

When fractions present, multiply both sides of the equation by the denominators.

Allows us to collect like terms more easily.

Immediately!

Expand out any brackets. Allows us to collect like terms more easily.

Once we we’ve eliminated fractions.

Factorise out new subject. Now only one occurrence of new subject in equation.

When terms involving subject all on one side of equation.

Collect terms involving new subject on one side of equation.

So that we can then factorise.

When new subject is free from fractions and brackets.

? ?

? ?

? ?

? ?

Using your solution to the previous question…

(Note: According to the Examiner’s Report, only 5% of students got full marks to this question)

Make a the subject of the formula:

Answer: ?

Another A* Question

?

?

?

Challenges

EdExcel Mock Paper

Q4

Q5

EdExcel Mar 2012:

Q3

EdExcel Nov 2012:

Q6

EdExcel May 2009:

Q2EdExcel Nov 2009:

Q7

Q8

EdExcel Nov 2009:

EdExcel Jun 2011:

Q9EdExcel Nov 2002:

O Level 1957:

Q10

Q1 ?

?

?

?

?

?

?

?

?

?

Questions

What tips would you give to someone answering the following questions?

Make a the subject of the formula: Make x the subject of the formula:

Final Tips?

𝑦−2𝑎2=𝑥

𝑥=𝑎+1𝑎−1

Make x the subject

𝑥=𝑎−1𝑎+1

𝑥=2𝑎−12𝑎+1

𝑥=1−𝑎1+𝑎

𝑥=(𝑎−𝑏𝑎 )2

𝑥=±√1−𝑎𝑏𝑥=±√ 𝑎−𝑏𝑎𝑥=±√ 𝑏−𝑎𝑎

Make x the subject

x = (a – b)/2a X = (ab – b)/2a X = (2a – b)/abX = (b – 2a)/b

Make x the subject

X = (b – a)/(ab – b2) X = (b2 + a)/(ab – a) X = (a – b)/(ab + b2)X = (b2 – a)/(ab – a)

Make x the subject

X = ba/c X = ac/b X = abcX = bc/a

Make x the subject

Further ExercisesIn all cases make the subject.

Q4

Q5

Q3

Q2

Q1

Q6

Q7

Q8

Q9

?

?

?

?

??

?

?

?