Rearranging Formulae

TL;DR

Rearranging formulae is all about isolating a specific variable by doing the same operation to both sides of an equation. You can think of it like "undoing" the operations around the variable you want to make the subject. The key is to work backwards through the order of operations, dealing with addition/subtraction first, then multiplication/division, and finally powers/roots.

1. The Mental Model

Imagine a formula is like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. Your goal is to get the variable you want all by itself on one side of the scale, and everything else on the other.

2. The Core Material

Rearranging formulae is a crucial skill in higher maths. It allows you to find an unknown value when you know all the others, or to understand the relationship between different parts of a formula.

What Does "Make X the Subject" Mean?

It means you want to get X by itself on one side of the equals sign, with no other terms or operations affecting it directly. For example, if you have A = L * W, and you want to make L the subject, you want to end up with L = A / W.

The Golden Rule: Do the Same to Both Sides

This is the absolute bedrock. Every single step you take must involve performing the same mathematical operation (add, subtract, multiply, divide, square, square root, etc.) to both sides of the equation.

Undoing Operations: The Reverse Order

Think about BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). When you're solving an equation for a number, you follow BIDMAS. When you're rearranging to isolate a variable, you effectively work backwards through BIDMAS:

1. Addition/Subtraction: Deal with any terms being added or subtracted away from the variable you want to isolate.
2. Multiplication/Division: Next, handle any terms multiplying or dividing the variable.
3. Indices/Roots: Finally, take care of any powers or roots applied directly to the variable.
4. Brackets: If the variable is inside a bracket, you often need to expand the bracket first, or treat the entire bracket as a single term until other operations are undone.

Here's a flow of how you might approach rearranging:

graph TD
    A["Start: Identify the variable to make subject"] --> B{Is the variable inside a bracket?};
    B -- Yes --> C["Expand bracket (if helpful) or treat as single term"];
    B -- No --> D["Deal with any terms added/subtracted (move them to other side)"];
    D --> E["Deal with any terms multiplying/dividing"];
    E --> F["Deal with any powers/roots"];
    F --> G["Isolate the variable"];
    G --> H["End: Variable is the subject"];

Worked Examples (Conceptual)

Let's look at some basic moves:

• Adding/Subtracting:
  If you have y = x + 5 and want to make x the subject:
  y - 5 = x + 5 - 5
y - 5 = x (or x = y - 5)

• Multiplying/Dividing:
  If you have A = LW and want to make W the subject:
  A / L = LW / L
A / L = W (or W = A / L)

• Powers/Roots:
  If you have A = πr² and want to make r the subject:
  First, get rid of π (it's multiplying r²):
  A / π = r²
  Then, get rid of the square by square rooting both sides:
  √(A / π) = √(r²)
√(A / π) = r (or r = √(A / π))

• Variables on Both Sides:
  Sometimes the variable you want appears on both sides. Collect all terms with that variable on one side, and everything else on the other. You might need to factorise it out.
  If 3x + 2 = 5x - 4 and you want x:
  2 + 4 = 5x - 3x (moved 3x and 4)
  6 = 2x
6 / 2 = x
3 = x

3. Worked Example

Let's make h the subject of the formula for the volume of a cone: V = (1/3)πr²h.

1. Identify the subject: We want h by itself.
2. Deal with the division/fraction: The (1/3) is dividing by 3. To undo this, multiply both sides by 3:
   3 * V = 3 * (1/3)πr²h
3V = πr²h
3. Deal with multiplication: π and r² are multiplying h. To undo this, divide both sides by πr²:
   3V / (πr²) = πr²h / (πr²)
3V / (πr²) = h
4. Final form: h = 3V / (πr²)

We've successfully made h the subject!

4. Key Takeaways

• Always perform the same operation on both sides of the equation to maintain balance.
• Work backwards through BIDMAS: deal with addition/subtraction first, then multiplication/division, then powers/roots.
• If the variable is inside a bracket, you might need to expand it or treat the whole bracket as a single term initially.
• When a variable appears multiple times, gather all its terms on one side and factorise if necessary.
• Your final answer should have the chosen variable completely isolated on one side of the equals sign.

Common Mistakes to Avoid

• Only operating on one side: This is the most common error and unbalances the equation.
• Incorrectly undoing an operation: Forgetting that subtraction undoes addition, division undoes multiplication, square root undoes squaring, etc.
• Dividing by zero or by a variable that could be zero: Be careful if you divide by a variable, as you assume it's not zero. (Less critical for basic rearranging but good to remember).
• Ignoring negative signs: Make sure to carry negative signs correctly when moving terms.
• Trying to deal with powers/roots before multiplication/division: Remember the reverse BIDMAS order.

5. Now Try It

Try to make x the subject of the formula y = (ax + b) / c.

What to do: Follow the steps we've discussed. First, think about what's "furthest away" from x in terms of operations, then work inwards.
What success looks like: You'll end up with an equation that starts x = ... where x is completely isolated. Your answer should be x = (cy - b) / a.