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:

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).