System of Two-by-Two Linear Equations Calculator

This calculator allows you to solve two simultaneous linear equations, with two variables, which are often times called, "two-by-two systems". These kind of 2x2 systems are very commonly used in Algebra, because they frequently appear in all kinds of applications, like when you try to solve word problems.

Typically, the variables used in a two-by-two linear system are called by default \(x\) and \(y\), but that is just a convention, as they could be \(u\) and \(v\) if you wish

So then, this is a two-by-two system:

\[x + 2y = 4\] \[2x - 2y = 2\]

the same way as this

\[2u - 2v = 1\] \[u - 3v = 2\]

is a two-by-two system. The important thing is that we have two linear equations with two variables (unknowns)

Methods for Solving a 2x2 linear systems

Fortunately, there are many ways you can use to solve two-by-two systems, and you have the benefit to choose which method to use. The most commonly used methods are:

• Graphing
• Substitution
• Elimination

The graphing method is based on, no surprise, graphing the two equations and trying to visually determine where these two lines intersect (if they intersect at all). This method is naturally limiting to approximations in most of the cases

The Substitution method is based upon the idea that one can solve for one variable in one of the equations, and then plug that into the other equation, to solve for the other variable. Often times this is convenient, because the structure of one of the equations may make it direct to solve for one variable. But this is not always the case, and this method is largely limited to the case of 2x2 systems

The Elimination method is based upon the idea that one can manipulate one or both equations to sum them or subtract them, so that one variable disappears. In a way, it is a more general way of using the substitution method

How to deal with larger systems of linear equations?

The three methods presented above really can only be efficiently used with 2x2 systems, as for larger systems the systems become much more complex and it may be even possible to use these methods

For 3x3 and large systems, it is best to use systematic approaches such as using Cramer's Method for general \(n \times n\) systems, or using Gaussian Elimination, which works regardless of the size of the system and whether the number of variables is the same as the number of equations.