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In mathematics, a system of equations is a collection of two or more equations that have a common set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. For example, in a system consisting of equations with two variables, like in our example, we are solving for the values of \(x\) and \(y\). Systems of equations come in three forms:

• Consistent systems - These have at least one solution and the equations intersect at a point.
• Inconsistent systems - These have no solution, meaning the graphs of the equations do not intersect.
• Dependent systems - The equations in the system are multiples of each other, representing the same line, resulting in infinitely many solutions.

Determining the type of system helps decide the best solving method. Analyzing the system before solving is crucial to understanding what kind of solutions to expect.

The elimination method, also known as the addition method, is a technique used to solve systems of equations. The main concept of this method is to eliminate one variable, making it easier to solve for the other. Here's how it typically works:

• First, align the equations so that corresponding variables line up vertically.
• Consider simplifying the equations if needed, such as by dividing all terms by a common factor.
• Make the coefficients of one of the variables the same or additive inverses, so when the two equations are added or subtracted, one variable is eliminated.
• Solve for the remaining variable.
• Substitute the found value back into one of the original equations to find the other variable.

In our example, we attempted to eliminate \(x\) by making the equations identical and subtracting them, which made both \(x\) and \(3y\) disappear, leaving us with a statement to analyze further.

A situation where a system of equations has no solution is referred to as an inconsistent system. This occurs when the equations represent parallel lines that never intersect. No matter how you try to solve for the variables, you'll end up with a nonsensical statement, like \(0 = -2\), indicating parallel lines.

This result usually stems from the equations having the same slope but different intercepts. Mathematically, this can often be identified either by getting an untrue statement after using methods such as substitution or elimination, or by graphing the equations to visually confirm that they do not meet. In this case, after simplifying the system and using the elimination method, we determined there was no point where both equations were satisfied, because the simplified equations had no solutions that worked concurrently.