Math - Solving Linear Equations

A linear equation is a mathematical statement with one or more unknown. Actually, every math puzzle you've ever solved since plussing one and one is an equation, it's just that you've never been told. 

 

Fundamentals:

When solving equations we usually use letters to symbolize what is unknown, usually x.  The simple equation two plus two would then be written as:
2 + 2 = x
Solving this we get:
2 + 2 = x
4 = x
This means that the unknown is four, and this is the answer to our puzzle.

So every puzzle is an equation, some easy, some harder.  As we are talking about linear equations, there are some more rules, making it easier for us.  Linear equations never multiply unknowns.  That means you will never see x*x in a linear equation.  Nor will you ever see x*y if you have more than one unknown.  Likewise is x/y never seen, that is, one unknown is never divided by another.

 

Solving:

Most equations aren't as easy as the one above.  The unknown can be anywhere in the puzzle, et can be added to, subratcted from, multiplied or divided.  Solving linear equations is all about getting things lined up like in the above example, with all the unknowns on one side, and all the number on the other.  It is then easy to solve most equations. 

An example:

Johnny has three apples, Jenny has some more.  Together they have eight apples.  This would be writtes as an equation as:
3 + x = 8

When dealing with equations you can do almost whatever you want, as long as you do it on both sides of the equals-sign.  If Peter has three apples, and Vicky has the same, and the both get four more, they still have an equal number.

An example:

3 = 3
3 + 4 = 3 + 4
7 = 7

This is true for almost anything, adding, subtracting, multiplying, dividing, and so on.  It is also true no matter if you use number, like the above, or unknowns.  This means we can now simplify and solve simple equations.

An example:

Johnny has three apples, Jenny has some more.  Together they have eight apples.
3 + x = 8
3 + x - 3 = 8 - 3
x + 3 - 3 = 8 - 3
x = 8 - 3
x = 5
Jenny has five apples.

Another example:

Greg has four apples, Andy has twice as many as Greg.  This means that the number of apples Andy has is unknown, and that Greg's four is half that unknown.
4 = x / 2
4 * 2 = x / 2 * 2
8 = 2 / 2 * x
8 = 1 * x
8 = x
Andy has eight apples.

You should now be able to solve simple linear equations with one unknown.