Solving Equations
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.
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
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 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
- 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
Start With | √(x/2) = 3 | |
Square both sides: | x/2^{ }= 3^{2} | |
3^{2} = 9: | x/2 = 9 | |
Multiply both sides by 2: | x = 18 |
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.
How To Check
Take the solution(s) and put them in the original equation to see if they really work.
Example: solve for x:
\frac{2x}{x − 3} + 3 = \frac{6}{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
Let us check:
\frac{2 × 3}{3 − 3} + 3 = \frac{6}{3 − 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 and 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).