Linear equations are the simplest type of equation and the easiest to understand. They are the usually the first type of equations you learn. There are not many types, as most can be solved by at most two methods.

Cancelling

Cancelling is to remove parts of the equation from both sides. There are a few rules, but I will not list them, as they are easy to work out once you get the hang of it. Let's take an example:

• Take 7x + 9 = 50.
• We do not want the 9, so we need to get rid of it.
• Think of the + 9 without the 7x. It really means 0 + 9.
• We know that 9 - 9 = 0, so to get rid of the +9 we -9.
• The -9 is called the inverse of the +9.
• We have to take away 9 from the other side too.
• We now have 7x = 41.

The next part is the most simple.

• If we have 7x = 41, we need the inverse of multiplying by 7.
• We know that if we multiply by z and divide by z, we get the original number.
• Therefore dividing is the inverse of multiplication.
• The inverse of 7x is then dividing by 7.
• If we do this to both sides, we are left with x = 41 / 7.
• That is the answer - you can work out 41 / 7 by yourself.

So what if x is on both sides of the equation?

If x is on both sides of the equation, such as 2x = x + 20, then the method of solution is a little harder (but not by much). Here's how we do it:

• Take 2x = x+20.
• Remember what inverses were?
• If we take x from both sides, we get 2x - x = 20.
• Now 2x is just x+x, so x+x - x = x.
• It's inverses again; -x is the inverse of +x.
• Therefore x = 20.
• This works: 20 * 2 = 20 + 20.

The idea of this method is to get x on one side of the equation so that you can solve it using the first method.

Inverses

Here are some inverses that are useful to know:

• + inverse is -
• * inverse is /
• ² inverse is square root
• ³ inverse is cube root

An inverse's inverse is the original, so the inverse of - is +.

These can be applied in algebra, so that the inverse of 2x is /x or /2, as both are possible.

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