Solving Equations
What's an Equation?
An equation says two things are equal. It will have an equals sign "=" like this:
| x | − | 2 | = | 4 |
That equations says:
what's on the left (x − 2) equals what's on the right (4)
So an equation is like a statement "this equals that"
What's 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 isn't a solution
- For x=9 we get "9−2=4" which is not true, so x=9 isn't a solution
- and so on
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
The above solution set is: {2, 3}
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's no "one perfect way" to solve all equations.
First follow the Golden Rule: to keep the balance, what we do to one side of the "=" we must also do to the other side.
A Useful Goal
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 right hand side.
Example: Solve 3x−6 = 9
Now we have x = something,
and a short calculation reveals x = 5
Variables on Both Sides
Sometimes we have variables on both sides of the equals sign. We can move them just like we move numbers.
Example: Solve 5x = 2x + 9
We want all the x's on one side.
Using Parentheses
If an equation has parentheses, we can use the Distributive Law to expand it first.
Example: Solve 3(x + 2) = 15
Like a Puzzle
In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and can't) 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
- Combine Like Terms
- Factoring
- 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)
Example: Solve √(x/2) = 3
And the more "tricks" and techniques you learn the better you will get.
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.
Let's multiply through by (x − 3):
2x + 3(x−3) = 6
Bring the 6 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
Which can be solved by having x=3
Let's check x=3 using the original question:
2 × 33 − 3 + 3 = 63 − 3
Hang On: 3 − 3 = 0
That means dividing by Zero!
And anyway, we said at the top that x≠3, so ...
x = 3 doesn't actually work, and so:
There's 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)