Solving Simple Systems of Linear Equations

What is a linear equation?

An algebraic equation is considered a linear equation simply because its graph is a straight line. You can check the post about graphing a simple linear function in the link that appears in parentheses (click here). In addition, a linear equation is one in which the variables involved have an exponent of 1. We can find a few examples of linear equations below.

As observed in the equations above, all the variables have 1 as an exponent. This makes them linear equations. The first two equations are linear equations with two unknowns, while the third equation is a linear equation with three unknowns. We will focus on the former in the remainder of this article.

What is a system of linear equations?

A system of linear equations is one that involves two or more linear equations with common variables as unknowns. When dealing with systems of linear equation, we are usually asked to find the solution of the system. The solution of the system is the value of the variables. 

In this article, we will only deal with systems of two linear equations with two unknowns. These are the simplest types of systems. When solving them, we often only apply two different methods. We will discuss these next.

Elimination of Variables

In this method, we seek to combine the given equations in such a way that we can eliminate one of the variables. From the combination of the equations, we can obtain a single equation with a single unknown that can be found by standard mathematical procedures. Subsequently, we will follow an example in which we apply this method.

Example 1:

Substitution of Variables

In this method, we solve either of the given equation for one of the unknowns. Then, the chosen equation is plugged into the other equation. Again, we obtain a single equation with a single unknown that can be found easily. 

Example 2:

Conclusion

As observed, the same solution was obtained by using the two different methods. There are more methods to solve linear equation. However, you will only need these two when solving simple systems of two linear equations with two unknowns. Needless to say, either method can applied. Your choice will depend on the type of system you have at hand. As you become more proficient, you will be able to choose the best for each particular system.