# 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)  equals  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

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 we can (and cannot) do.

Here are some things we can do:

### Example: Solve √(x/2) = 3

Square both sides:x/2= 32
Calculate 32 = 9:x/2 = 9
Multiply both sides by 2:x = 18

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 ...

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:

2xx − 3 + 3 = 6x − 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

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0
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)

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