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.
|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
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
- 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.
- 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).