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:

Example: Solve √(x/2) = 3

Start With   √(x/2) = 3
Square both sides:   x/2 = 32
32 = 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.

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.

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).