Solving Equations

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 ?

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 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
Calculate 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.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

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 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 × 3 3 − 3 + 3  =   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!

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