🎯 What You're About to Master
Rearranging formulas (also called "changing the subject") means taking a formula like y = mx + c and rewriting it to solve for a different variable. This skill is essential not just in maths exams, but everywhere: physics, engineering, chemistry. You'll use it to solve real problems and to make calculations easier.
Why this matters
In science and engineering, you often have a formula but need to find a different variable. For example, you might know the area of a circle (A = πr²) but need to find the radius (r). Rearranging formulas is the only way to do that. This question appears on almost every higher maths paper.
📖 Understanding Formulas
What Is a Formula?
A formula is a rule showing relationships between quantities
• It uses letters (variables) to represent quantities
• It always contains an equals sign (=)
• It shows how one quantity depends on others
Common Formulas You Should Know
Straight line equation: y = mx + c
Area of trapezium: A = (a + b)h ÷ 2
Pythagoras' theorem: a² + b² = c²
Kinetic energy: E = ½mv²
What Is the Subject?
The subject is the letter that's alone on one side of the formula
In y = mx + c, the subject is y (it's on its own on the left)
Changing the subject means making a different letter the subject by rearranging.
🔄 Simple Rearranging (Subject Appears Once)
The Core Strategy
1
Remove fractions first — multiply by the lowest common denominator
2
Use inverse operations — undo operations in reverse order (add/subtract, then multiply/divide)
3
Check your answer — substitute back into the original formula
What Are Inverse Operations?
Operation
+ (add)
− (subtract)
× (multiply)
÷ (divide)
Inverse (Undo)
− (subtract)
+ (add)
÷ (divide)
× (multiply)
Example 1: Make x the subject of (5x + 6)/2 = y
Step 1: Remove the fraction
Multiply both sides by 2:
5x + 6 = 2y
Step 2: Get 5x on its own
Subtract 6 from both sides:
5x = 2y − 6
Step 3: Get x on its own
Divide both sides by 5:
x = (2y − 6)/5
Step 4: Check alternative forms
These are all equivalent:
x = (2y − 6)/5 or x = 2y/5 − 6/5 or x = 2(y − 3)/5
Deciding Whether to Expand Brackets
If the variable is inside brackets, you have two choices:
Option 1: Expand first
3(1 + x) = y
3 + 3x = y
3x = y − 3
x = (y − 3)/3
Option 2: Divide first
3(1 + x) = y
1 + x = y/3
x = y/3 − 1
Both answers are equivalent! The form (y − 3)/3 equals y/3 − 1. Use whichever method feels easier.
Key insight
If the variable is NOT inside brackets, don't expand. Just divide straight away. For example, to make x the subject of (1 + k)x = y, divide both sides by (1 + k): x = y/(1 + k). No need to expand the bracket.
1
Make m the subject of: 4m + 5x = 3
Dealing With Fractions in Fractions
If you end up with something like x = (3/t)/2, you have two methods:
Example 2: Simplify (3/t)/2
Method 1: Convert to division
x = 3/t ÷ 2
= 3/t × 1/2
= 3/(2t)
Method 2: Multiply top and bottom
⭐
The Key Insight: Factorise!