The one rule behind everything
A formula is a balance. Whatever you do to one side of the equals sign, you must do to the other. Every technique for rearranging formulas is just this rule applied with a purpose: get the variable you want alone on one side.
Work in reverse order of operations. If the variable has been multiplied, then added to something, undo the addition first, then the multiplication. You are peeling layers off, outermost first.
Example 1: a simple product
Ohm’s law is usually written V = I × R. To find resistance, R, divide both sides by I:
R = V / I
Check with numbers: if V = 12 and I = 3, then R = 12 / 3 = 4. Substituting back, 3 × 4 = 12, which matches V. The same idea gives I = V / R.
Example 2: a straight line
The equation of a straight line is y = m x + b. Solve it for x.
- Subtract b from both sides: y − b = m x
- Divide both sides by m: x = (y − b) / m
Check with m = 2, b = 3 and x = 4. Then y = 2 × 4 + 3 = 11. Using the rearranged form, x = (11 − 3) / 2 = 8 / 2 = 4. It matches.
A common slip is dividing only part of the expression. Dividing by m means the whole of (y − b) is divided, which is why brackets matter.
Example 3: a square and a root
Power in a resistor can be written P = V² / R. Solve for V.
- Multiply both sides by R: P R = V²
- Take the square root of both sides: V = √(P R)
Check with V = 6 and R = 4. Then P = 36 / 4 = 9. Using the rearranged form, V = √(9 × 4) = √36 = 6. It matches.
Square roots have a positive and a negative solution in pure algebra, since (−6)² is also 36. In physical problems you usually keep only the root that makes sense, such as a positive voltage magnitude. Decide this from the context, not by habit.
Example 4: kinetic energy
Kinetic energy is E = ½ m v². Solve for v.
- Multiply both sides by 2: 2E = m v²
- Divide both sides by m: 2E / m = v²
- Take the square root: v = √(2E / m)
Check with m = 2 and v = 3. Then E = ½ × 2 × 9 = 9. Using the rearranged form, v = √(2 × 9 / 2) = √9 = 3. It matches.
Example 5: reciprocals
Two resistors in parallel combine as 1/R = 1/R₁ + 1/R₂. Suppose you know R and R₁ and want R₂.
- Subtract 1/R₁ from both sides: 1/R₂ = 1/R − 1/R₁
- Combine the right side over a common denominator if needed, then flip the result to get R₂.
Check with R₁ = 6 and R₂ = 3. The combined value is found from 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2, so R = 2. Now rearrange: 1/R₂ = 1/2 − 1/6 = 3/6 − 1/6 = 2/6 = 1/3, so R₂ = 3. It matches.
Flip the fraction only when one side is a single fraction. Flipping term by term is a frequent error.
Example 6: an offset and a scale factor
The Fahrenheit and Celsius scales are related by F = (9/5) C + 32. To solve for C:
- Subtract 32: F − 32 = (9/5) C
- Multiply both sides by 5/9: C = (5/9)(F − 32)
Check with F = 212: C = (5/9) × 180 = 100. Water boils at 100 °C at standard atmospheric pressure, so this is the expected result.
Habits that prevent mistakes
- Write each step on its own line. Doing two operations at once is where signs get lost.
- Keep brackets around anything being divided or multiplied.
- Substitute simple numbers at the end. Choose values that make the arithmetic easy, as above. If both versions of the formula agree, you can be fairly confident.
- Track units. The rearranged formula must have units that make sense, which is the idea behind dimensional analysis. In V = √(P R), watts times ohms gives volts squared, so the root is in volts.
- Rearrange first, substitute later. Working with symbols, then putting numbers in once, usually creates fewer rounding errors and leaves a reusable formula, for instance in engineering spreadsheets.
When a variable appears twice
Sometimes the variable you want appears in more than one place, as in a x + b = c x + d. Gather all the terms containing x on one side and everything else on the other:
a x − c x = d − b, so x (a − c) = d − b, so x = (d − b) / (a − c)
Factorising is what lets you finish. Check with a = 5, b = 1, c = 2, d = 10. The formula gives x = (10 − 1) / (5 − 2) = 9 / 3 = 3. Left side: 5 × 3 + 1 = 16. Right side: 2 × 3 + 10 = 16. They agree.
Last check
After rearranging, ask whether the result behaves sensibly. In R = V / I, if the current doubles for the same voltage, the resistance should halve, and it does. Quick sanity checks like this catch many errors. These examples use simple illustrative numbers; for real design work, follow the methods and standards that apply to your field.