Rearranging and Applying Formulas

TL;DR

Rearranging formulas means isolating a different variable, treating other variables like numbers. You use inverse operations to "undo" what's been done to the variable you want to isolate. This skill lets you solve for any part of a relationship, not just the original output.

1. The Mental Model

Think of a formula like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. Your goal is to move everything except the variable you want to solve for to the other side of the equals sign.

2. The Core Material

Formulas are just equations that show how different quantities relate. Rearranging them means changing which variable is "alone" on one side of the equals sign. This is super useful because sometimes you know the result and want to find one of the inputs, or you might need a formula in a specific form for another calculation.

The Golden Rule: Inverse Operations

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To move a term to the other side, you use the inverse operation.

• If something is added (+), you subtract it from both sides.
• If something is subtracted (-), you add it to both sides.
• If something is multiplied (*), you divide by it on both sides.
• If something is divided (/), you multiply by it on both sides.
• If a variable is squared (x²), you take the square root (√) of both sides.
• If a variable is under a square root (√x), you square both sides.

Always perform operations in reverse order of operations (PEMDAS/BODMAS) when undoing an equation. This means you generally undo addition/subtraction first, then multiplication/division, and finally powers/roots.

Step-by-Step Process

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graph TD
    A["Identify variable to isolate"] --> B["Locate variable in formula"]
    B --> C["Undo addition/subtraction (farthest from variable)"]
    C --> D["Undo multiplication/division"]
    D --> E["Undo powers/roots"]
    E --> F["Check: Is variable isolated?"]
    F --> |No| C
    F --> |Yes| G["Done!"]

Let's look at an example:

Original Formula: V = LWH (Volume of a rectangular prism = Length * Width * Height)

Goal: Isolate H (Height)

1. Identify variable to isolate: H
2. Locate variable: H is being multiplied by L and W.
3. Undo operations: Since H is multiplied by LW, the inverse operation is to divide by LW.
   V / (LW) = (LWH) / (LW)
4. Simplify: V / (LW) = H
5. Result: H = V / (LW)

Formulas with Addition/Subtraction

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Let's try a slightly more complex one:

Original Formula: P = 2L + 2W (Perimeter of a rectangle = 2 * Length + 2 * Width)

Goal: Isolate L (Length)

1. Identify variable to isolate: L
2. Locate variable: L is multiplied by 2, and then 2W is added to that product.
3. Undo addition/subtraction first: 2W is added to 2L. Subtract 2W from both sides.
   P - 2W = 2L + 2W - 2W
P - 2W = 2L
4. Undo multiplication/division: L is multiplied by 2. Divide both sides by 2.
   (P - 2W) / 2 = (2L) / 2
5. Simplify: (P - 2W) / 2 = L
6. Result: L = (P - 2W) / 2

3. Worked Example

Let's use the formula for converting Celsius to Fahrenheit: F = (9/5)C + 32.

Suppose you have a temperature of 77°F and you want to know what it is in Celsius. You need to rearrange the formula to solve for C.

Original Formula: F = (9/5)C + 32
Given: F = 77
Goal: Isolate C

1. Undo addition/subtraction: 32 is added to (9/5)C. Subtract 32 from both sides.
   F - 32 = (9/5)C + 32 - 32
F - 32 = (9/5)C

2. Undo multiplication/division: C is multiplied by 9/5. To undo this, multiply both sides by the reciprocal, which is 5/9.
   (5/9)(F - 32) = (5/9)(9/5)C
(5/9)(F - 32) = C

3. Rearranged Formula: C = (5/9)(F - 32)

4. Apply the value: Now substitute F = 77 into the rearranged formula.
   C = (5/9)(77 - 32)
C = (5/9)(45)
C = (5 * 45) / 9
C = 225 / 9
C = 25

So, 77°F is 25°C.

4. Key Takeaways

• Rearranging formulas means isolating a different variable on one side of the equals sign.
• Always perform the same operation to both sides of the equation to keep it balanced.
• Use inverse operations: addition undoes subtraction, multiplication undoes division, square roots undo squares.
• Generally, undo addition/subtraction terms first, then multiplication/division terms.
• Treat other variables in the formula as if they were known numbers.
• Rearranging formulas is crucial for solving for any unknown in a given relationship.
• Practise makes perfect; the more you do, the more natural it becomes.

Common Mistakes to Avoid:
- Forgetting to apply an operation to both sides of the equation.
- Incorrectly performing inverse operations (e.g., adding when you should subtract).
- Trying to move a term that's inside a multiplication/division before moving the multiplication/division itself.
- Mixing up the order of undoing operations (e.g., dividing before subtracting when it's ax + b).

5. Now Try It

Rearrange the formula for the area of a trapezoid, A = (1/2)h(b1 + b2), to solve for h (the height). Then, calculate h if A = 36 cm², b1 = 8 cm, and b2 = 10 cm. What you should get is a clear, step-by-step rearrangement of the formula, followed by the correct numerical answer for h with units.