Re-Arrange Any Algebraic Equation with One Simple Rule

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The harsh truth is that a lot of people don’t like math, and if there is one element of math that puts people off the most, it’s algebra. The mere mention of the word is enough to raise a collective groan from every student from seventh grade and up. But if you’re hoping to get into a good college or just get good grades, you’ll have to get to grips with it. The good news is that it isn’t actually as bad as you think. Once you get used to the fact that you’re using letters and symbols to stand-in for numbers, there’s really one major rule you have to master: Do the same thing to both sides of the equation when re-arranging.

The Most Important Algebra Rule

The most important rule for algebra is: If you do something to one side of an equation, you have to do it to the other side too.

An equation basically says “the stuff on the left hand side of the equals sign has the same value as the stuff on the right hand side of it,” like a balanced set of scales with equal weights on both sides. If you want to keep everything equal, anything you do needs to be done to both sides.

Looking at a basic example using numbers really drives this home.

2 × 8 = 16

This is obviously true: Two lots of eight are indeed equal to 16. If you multiply both sides by two again, to give:

2 × 2 × 8 = 2 × 16

Then both sides are still equal. Because 2 × 2 × 8 = 32 and 2 × 16 = 32 as well. If you did this to one side only, like this:

2 × 2 × 8 = 16

You’d actually be saying 32 = 16, which is clearly wrong!

By changing the numbers to letters, you get an algebraic version of the same thing.

x × y = z

Or simply

xy = z

It doesn’t matter that you don’t know what x, y or z mean; on the basis of this basic rule you know that all of these equations are true as well:

2xy = 2z \\ xy / 4 = z/4 \\ xy + t = z + t

In each case, exactly the same thing has been done to both sides. The first multiplies both sides by two, the second divides both sides by four, and the third adds another unknown term, t, onto both sides.

Learning the Inverse Operations

This basic rule is really all you need to re-arrange equations, along with the rules for which operations cancel out which others. These are called “inverse” operations. For example, the inverse of adding is subtracting. So if you have x + 23 = 26, you can subtract 23 from both sides to remove the “+ 23” part on the left:

\begin{aligned} x + 23 −23 &= 26 − 23 \\ x &= 3 \end{aligned}

Likewise, you could cancel out subtraction using addition. Here is a list of some common operations and their inverse (which all apply the opposite way around too):

    • is cancelled

    by –

  • × is cancelled by

÷

  • √ is cancelled by 2
  • ∛ is cancelled by 3

Others include the fact that e raised to a power can be called out using the “ln” operation and vice-versa.

Practice at Re-Arranging Equations

With this in mind, you can re-arrange pretty much any equation you come across. The goal when you re-arrange an equation is usually isolating a specific term. For example, if you have the equation for the area of a circle:

A = πr^2

You might want an equation for r instead. So you cancel the multiplication of r2 by pi by dividing by pi. Remember that you have to do the same thing to both sides:

{A \above{1pt} π} = {πr^2 \above{1pt} π}

So this leaves:

{A \above{1pt} π} = r^2

Finally, to remove the squared symbol on the r, you need to take the square root of both sides:

\sqrt{A \above{1pt} π} = \sqrt {r^2}

Which (turning it around) leaves:

r=\sqrt{A \above{1pt} π}

Here’s another example you can practice with. Imagine you have this equation:

v = u + at

And you want an equation for a. What do you have to do? Try it before reading on, and remember that what you do to one side you have to do to the whole of the other side.

So starting with

v = u + at

You can subtract u from both sides (and reverse the equation) to get:

at = v – u

Finally, get your equation for a by dividing by the t:

a = {v \; – \; u \above{1pt} t}

Note that you can’t just divide u by t in the last step: you have to divide the whole of the right side by t.

References

About the Author

Lee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts into simple, digestible language. He's written about science for several websites including eHow UK and WiseGeek, mainly covering physics and astronomy. He was also a science blogger for Elements Behavioral Health's blog network for five years. He studied physics at the Open University and graduated in 2018.

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