Linear equations are a specific kind of equation that meets certain limitations. Specifically, linear equations are those where every variable (if there are any variables at all) is raised only to the first power (see here for a guide to powers/exponents). Furthermore, those variables must not be multiplied by any other variables.
Here are some examples of linear equations:
Here are some examples of equations that are not linear:
In subsequent posts, we will discuss two methods for solving linear equations: elimination and substitution. We will also talk about systems of linear inequalities, which occur when we replace those equality signs with inequalities like or .
For now, see if you can identify whether the following equations are linear or not:
Identify whether the following equations are linear or not
No, since is not a single variable raised to the first power.
Yes, since involves just a single variable raised to the first power. It is multiplied by a constant (namely 4), but that is fine.
Technically no, since is not a single variable raised to the first power. But we can see that since no matter what the value of and are, our equation has the same solutions as the linear equation
No, since is not a linear term.
Technically no, since is not a linear term. But provided we can multiply both sides by it in order to get which is linear. So again, our equation has the same solutions as a linear equation.
No, and our earlier trick for problem 5 will not even work here since is still not a linear equation.
Now, systems of linear equations are just groups of 1 or more linear equations. The solutions to the system are all the different ways of assigning values to each variable such that every equation in the group is true. Note that every solution must specify the value for each variable in our system. So, for example, in the following system:
The response is not a solution. However, is a solution. And in general, we can give our solutions in the form of ordered pairs. So we would write (2, 3) is a solution, where the first coordinate corresponds to the value of and the second coordinate corresponds to the value of .
The reason for requiring that solutions specify values for all variables (not just some) is that sometimes a certain value for one variable can be a solution and sometimes not, depending on the values of the other variables. Consider:
Now, this system has many solutions. For example, is a solution. But we cannot simply say that is a solution, for if , then that is no solution at all (since )! Thus, we require that a solution for a system of equations specify values for every variable in the system.
In our next post, we discuss how to solve systems of linear equations via elimination.