Linear Equations

What are linear equations? Well, they often look something like this: p + 4 = 5. Or 3x = 2. These are simple and basic linear equations. We want to point out something very important right off the bat: it’s possible to solve linear equations! Yes, you heard this correctly. In-fact, solving linear equations is one of our main goals. This allows us to determine the value of a given variable. But before we get ahead of ourselves, I think we should define some important key terms. 

Key Terms

In most linear equations you will have a variable, coefficient and constant. 

Variable – x, y, or z (for example)

Coefficient – the number before the variable 

Constant  – a value which is unchanging 

Examples

3p = 6

Ok. So what’s the variable here? The variable is clearly p. It is a letter used to represent a value which we don’t yet know. It’s really cool to think of the variable as simply a representation of something. We can make it like a game. We are trying to locate and uncover the value of the variable. In this instance, we are given the coefficient pretty clearly. We can see that the number before the variable (p) is 3. Therefore, the coefficient is 3. Sometimes the coefficient will be hidden (for example if the equation was p = 6 instead of 3p = 6). In the case of p = 6, the coefficient is just 1. It’s there. It’s just not written. 

To solve linear equations we simply use the arithmetic that we already know. In order to effectively solve linear equations we can use:

  • Addition & Subtraction
  • Multiplication and Division

3p = 6

Our first step here would be divide by 3. This leaves us with:

3p/3 = 6/3

On the left side the 3’s cancel. And that leaves us with: 

p = 2

That’s the basics of solving linear equations. Things can get much more challenging, however. It’s common for questions to use multiple variables. It’s also common for some coefficients and constants to be negative. This often throws students off. But it’s important to remember that we can always multiple by a negative number. 

Important Rules

That leaves us with some of the important rules to remember when you are trying to solve linear equations. 

Anything you do to one side you must do to another.

Anything & everything! Remember this important rule. Recall the example that we showed above. We divided by 3 on both of the sides. We can also do things such as multiple by -1. But remember, we must multiple by -1 on both sides! 

This is the most important rule to remember for all of linear equations! Remember this and you are bound to improve your skills at solving and understanding linear equations.