# Solving Equations

## What is an Equation?

An equation says that two things are equal. It will have an equals sign "=" like this:

x |
− |
2 |
= |
4 |

That equations says: **what is on the left (x − 2) is equal to what is on the right (4)**

So an equation is like a **statement** "*this* equals *that*"

## What is a Solution?

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

### Example: x − 2 = 4

When we put 6 in place of x we get:

6 − 2 = 4

which is **true**

So x = 6 is a solution.

How about other values for x ?

- For x=5 we get "5−2=4" which is
**not true**, so**x=5 is not a solution**. - For x=9 we get "9−2=4" which is
**not true**, so**x=9 is not a solution**. - etc

In this case x = 6 is the only solution.

You might like to practice solving some animated equations.

## More Than One Solution

There can be **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 also **true**

So the solutions are:

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

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

The above solution set is: {2, 3}

## Solutions Everywhere!

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

### Example: **sin(−θ) = −sin(θ)** is one of the Trigonometric Identities

Let's try θ = 30°:

**sin(−30°) = −0.5** and

**−sin(30°) = −0.5**

So it is **true** for θ = 30°

Let's try θ = 90°:

**sin(−90°) = −1** and

**−sin(90°) = −1**

So it is also **true** for θ = 90°

Is it true for **all values of θ**? Try some values for yourself!

## How to Solve an Equation

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

### A Useful Goal

But we often get success when **our goal** is to end up with:

**x** = *something*

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

### Example: Solve 3x−6 = 9

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 we can (and cannot) do.

Here are some things we can do:

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

### Example: Solve √(x/2) = 3

^{ }= 3

^{2}

^{2}= 9:x/2 = 9

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

## Special Equations

There are special ways of solving some types of equations. Learn how to ...

- solve Quadratic Equations
- solve Radical Equations
- solve Equations with Sine, Cosine and Tangent

## Check Your Solutions

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

### How To Check

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

### Example: solve for x:

\frac{2x}{x − 3} + 3 = \frac{6}{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

Let us check:

\frac{2 × 3}{3 − 3} + 3 = \frac{6}{3 − 3}

**Hang On!
That means Dividing by Zero! **

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 us possible solutions, they need to be checked!

## 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)