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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) is equal to 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
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:
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
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:
- Add or Subtract the same value from both sides
- Clear out any fractions by Multiplying every term by the bottom parts
- Divide every term by the same nonzero value
- Expanding (the opposite of factoring) may also help
- Recognizing a pattern, such as the difference of squares
- Sometimes we can apply a function to both sides (e.g. square both sides)
Math Problem Solving Answer
Example: Solve √(x/2) = 3
Math Problem Solving Questions
And the more 'tricks' and techniques you learn the better you will get. Poetry unitmrs. parkers 6th grade l.a. class.
Special Equations
There are special ways of solving some types of equations. Learn how to ..
- solve Quadratic Equations
- solve Radical Equations
- solve Equations with Sine, Cosine and Tangent
Check Your Solutions
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. Poser 11 free download.
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
Let us check:
2 × 33 − 3 + 3 = 63 − 3
Hang On!
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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