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Date: 13-2-2019
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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:
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x - 2 = 4 |
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If we put 6 in place of x we get: |
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6 - 2 = 4 |
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, which is true |
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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 |
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3x-6 = 9 |
Add 6 to both sides: |
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3x = 9+6 |
Divide by 3: |
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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:
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2x |
+ 3 = |
6 |
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(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:
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2·3 |
+ 3 = |
6 |
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Hang On! That would mean |
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
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