Believe it or not, algebra exists for a reason other than lowering a high school student's grade point average. Systems of linear equations, or a set of equations with two or more variables, are an essential part of finding solutions with only limited information, which happens to be exactly what algebra is. As a required part of any algebra student's life, it is best to understand how they work, not only so an acceptable grade is received, but also so one day the systems can be used to actually find desired information with ease. There are three main methods of defining a system of linear equations.
One way is called a consistent, independent solution. This essentially means that the system has one unique, definite solution. In this situation on a graph, a set of two equations and two variables would be solved as one single point where two lines intersect. It is much the same with three variables and three equations. The only difference is that the point is an intersection of three planes instead of two lines. Additionally, there are situations where a system of linear equations could be described as consistent, dependent.
These systems of linear equations have an infinite number of solutions where a general solution is used to substitute one or two variables for one other selected variable, and solves the other unknown variable or variables in terms of that selected one. Graphically when this system of linear equations is solved for two equations and two variables, the result is lines that coincide, or lay on top of each other, making any point on that line true for the system. A system with three equations and three variables would yield an answer that shows the three planes intersecting on a line or overlaying each other. When the system yields three planes intersecting on a line, all points on the line would make the systems true, and when planes coincide, all points on the coincidental planes would be correct when placed in the system for the variables. A final case exists with systems of linear equations where the system can be classified as inconsistent, independent. This classification means that the system has no solution or that there is not a single point where all the lines or planes intersect.
In this case, with two variables and two equations, the lines are parallel, never intersecting at one point. When the system is three variables and three equations, no solution means that the three planes do not touch at all, or there is not one point where all three planes intersect. Systems of linear equations with two or three variables have specific rules and steps to solving the variables. As a great way of finding additional information when only presented with limited sources, knowing how to decipher these systems of linear equations can, indeed, be very useful.
Algebra can be a difficult part of high school, but understanding the essential parts of algebra and understanding what it is about are the keys to success.