# Solving equations

First, simplify on boths sides. On the left side you can add and . Then you get the equation:

Next, you have to rearrange the equation in such a way that x is on the left side and numbers on the right side. Since we don't like the x on the right side, we substract x on both sides. are left on the left side.

Now, we have to get the number on the other side. So we add on bothe sides. Since , we get

Now, we divide both sides by the number in front of the x:

The equation is solved now; is a solution of it.

In the exact same manner you can always proceed: First, simplify both sides of the equation as far as possible. Then, simplify with equivalence transformations. Subtract a number cleverly on both sides Finally, there should be a multiple of the variables on one sode and a number on the other side. You divide by the number in front of the variable and the equation is solved.

Solution set: {} |

First, an example of an equation with an infinite number of solutions:

Solution set: R |

You see that you end up with the same numbers on both side. It's obviously a true statement for any value of x (there is no x in this equation anymore). Thus, we see that an equation can have an infinite number of solutiosn.

What does it mean when an equation has got an infinite number of solutions? You can try it out: Take any value for x (e.g. , both sides will be the same. It works with any value for x. The reason is, that the terms on both sides are equivalent, i.e. terms with the same solution with any value for x.

The other special case is an equation with no solution:

Solution set: {} |

We see that there is no x in the equation after rearranging and that the equation is obviously false. This is due to the original equation having no solution.