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.
In this case x = 6 is the only solution.
More Than One Solution
But we 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 we 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 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
|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 we can (and cannot) do.
Here are some things we 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 we can apply a function to both sides (e.g. square both sides).
- Recognizing a pattern, such as 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 means
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 and found it wasn't allowed!
This gives us a moral lesson:
"Solving" only gives us possible solutions, they need to be checked!
How To Check
Take the 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).