Linear Systems Summary
If you have two lines on the same graph, one of three things will happen. First, they could turn out to be the same line, so they intersect at an infinite number of points. Second, they could turn out to be parallel lines, so they never intersect at any point. In every other case, the two lines intersect at exactly one point.
Graphing two equations on the same graph is the visual way of thinking about a system of two linear equations. For example, if we have the equations 3x + 6y = -12 and 3x – y = 9 together, we call that a system. Once we have a system, we can try to find the ordered pair that solves both equations, thus solving the system.
Sometimes you won’t be able to solve a system for an ordered pair. This happens for two possible reasons. The first reason is that there is either an infinite number of solutions, and this occurs when the two equations are actually the same line. The second reason could be that the equations are equations of two parallel lines, and thus never intersect, meaning no point solves both equations at the same time.
There are three basic approaches generally taught in any Algebra 1 class, and then a fourth special way we’re going to show you that isn’t usually taught. The three traditional approaches are solving by substitution, graphing and elimination. Our special fourth way allows you to apply critical thinking to the system of equations without getting bogged down in all of the technical aspects of Algebra 1.
It’s advised that you work hard on the idea of substitution because it’s an idea that you’ll use a lot in later topics in Algebra 1. If you plan to go past Algebra 1 in math courses, then understanding the basic idea of how substitution works is a must. You should already understand how to solve a system of equations by graphing since it’s just a simple extension of the graphing you’ve done earlier of one line at a time. Solving by elimination is a simple shortcut, and while it’s easy to understand, it’s not very practical to spend a lot of time on since it only comes up in some cherry-picked situations and doesn’t happen very often naturally.
Nature of the Solutions – There are only three ways a linear system can work out.
Solving by Graphing – A visual way to solve a system of two equations.
Solving by Substitution – An algebraic method for solving systems of equations.
Solving by Elimination – A neat trick that makes solving systems algebraically very easy in some specific situations.
A Simple Approach to Solving Systems of Equations – This shows you the easiest way to solve a system of equations algebraically with step-by-step instructions.