Rearranging equations

Rearranging equations involves taking an equation and changing its form to get one of the variables on its own.

The key thing to remember when rearranging equations is that whatever you do to one side of the equation, you have to do to the other side as well. This keeps the equation balanced.

The easiest way to rearrange equations is to think about what happens to the variable you are trying to find, and then tell that story in reverse.

Rearranging simple linear equations

Let’s try some examples

Example: Make x the subject of the equation y = 2x + 3.

• Write down the equation:
  • y = 2x + 3
• ‘Tell a story’ of what has happened to x:
  • We started with x
  • We multiplied by 2.
  • We added 3.
  • We ended up with y.
• Tell the story in reverse:
  • We start with y.
  • We subtract 3.
  • We divide by 2.
  • We end up with x.
• Subtract 3 from both sides:
  • y - 3 = 2x
• Divide both sides by 2:
  • \frac{y - 3}{2} = x
• Write the final answer:
  • x = \frac{y - 3}{2}

Example: Make V the subject of the equation P = IV.

• Write down the equation:

  • P = IV

• ‘Tell a story’ of what has happened to V:

  • We started with V.
  • We multiplied by I.
  • We ended up with P.

• Tell the story in reverse:

  • We start with P.
  • We divide by I.
  • We end up with V.

• Divide both sides by I:

  • \frac{P}{I} = V

• Write the final answer:

  • V = \frac{P}{I}

  More complex, non-linear rearrangements

  Example: Make v the subject of the equation E = \frac{1}{2}mv^2.

• Write down the equation:

  • E = \frac{1}{2}mv^2

• ‘Tell a story’ of what has happened to v:

  • We started with v.
  • We squared it.
  • We multiplied by m.
  • We multiplied by \frac{1}{2}.
  • We ended up with E.

• Tell the story in reverse:

  • We start with E.
  • We multiply by 2.
  • We divide by m.
  • We take the square root.
  • We end up with v.

• Multiply both sides by 2:

  • 2E = mv^2

• Divide both sides by m:

  • \frac{2E}{m} = v^2

• Take the square root of both sides:

  • v = \sqrt{\frac{2E}{m}}

• Write the final answer:

  • v = \sqrt{\frac{2E}{m}}

Example: Make q the subject of the equation V = \frac{E}{2q}.

• Write down the equation:
  • V = \frac{E}{2q}
• ‘Tell a story’ of what has happened to q:
  • We started with q.
  • We multiplied by 2.
  • We divided E by that result.
  • We ended up with V.
• Tell the story in reverse:
  • We start with V.
  • We divide E by V.
  • We divide by 2.
  • We end up with q.
• Multiply both sides by 2q:
  • 2qV = E
• Divide both sides by V:
  • 2q = \frac{E}{V}
• Divide both sides by 2:
  • q = \frac{E}{2V}
• Write the final answer:
  • q = \frac{E}{2V}

What if the term we want is on both sides?

TODO

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