# Solving Equations

## What is a Solution?

A Solution is a value you can put in place of a variable (such as *x*) that would make the equation **true**.

### Example:

x - 2 = 4 | ||||

If we put 6 in place of x we get: | 6 - 2 = 4 | , which is true |
||

So x = 6 is a solution |

Note: try another value for x. Say x=5: you get 5-2=4 which is **not true**, so **x=5 is not a solution**.

## More Than One Solution

You can have more than one solution.

### Example: (x-3)(x-2) = 0

When x is 3 we get:

(3-3)(3-2) = 0 × 1 = 0

which is **true**

And when x is 2 we get:

(2-3)(2-2) = (-1) × 0 = 0

which is **true**

So the solutions are:

x = **3**, or x = **2**

When you gather all solutions together it is called a **Solution Set**

## Solutions Everywhere!

Some equations are true for all allowed values and are then called **Identities**

Example: this is one of the trigonometric identities:

**tan(θ) = sin(θ)/****cos(θ)**

## How to Solve an Equation

There is no "one perfect way" to solve all equations.

### A Useful Goal

But you will often get success if **your goal** is to end up with:

**x** = *something*

In other words, you want to move everything except "x" (or whatever name your variable has) over to the right hand side.

### Example: Solve 3x-6 = 9

Start With | 3x-6 = 9 | |

Add 6 to both sides: | 3x = 9+6 | |

Divide by 3: | x = (9+6)/3 |

Now we have **x = something**,

and a short calculation reveals that **x = 5**

## Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things you can (and cannot) do.

Here are some things you can do:

- Clear out any fractions by Multiplying every term by the bottom parts.
- Add or Subtract the same value from both sides.
- Divide every term by the same nonzero value.
- Combine Like Terms
- Factoring
- Expanding (the opposite of factoring) may also help
- Sometimes you can apply a function to both sides (e.g. square both sides).
- Recognizing a pattern you have seen before, like the difference of squares

And the more "tricks" and techniques you learn the better you will get.

## Check Your Solutions

You should always check that your "solution" really **is** a solution.

### Example: solve for x:

2x | + 3 = | 6 | (x≠3) | ||

x - 3 | x - 3 |

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x - 3):

2x + 3(x-3) = 6

Bring the 6 to the left:

2x + 3(x-3) - 6 = 0

Expand and solve:

2x + 3x - 9 - 6 = 0

5x - 15 = 0

5(x - 3) = 0

x - 3 = 0

That can be solved by having x=3, so let us check:

2·3 | + 3 = | 6 | Hang On! That would mean Dividing by Zero! |
||

3 - 3 | 3 - 3 |

And anyway, we said at the top that x≠3, so ...

x = 3 does not actually work, and so:

There is **No** Solution!

That was interesting ... we **thought** we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives you possible solutions, they need to be checked!

## How To Check

Take your solution(s) and put them in the **original equation** to see if they really work.

## Tips

- Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
- Show all the steps, so it can be checked later (by you or someone else).