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In algebra, a system of equations, also known as an algebraic system, is a set of two or more equations that involve the same set of variables. The objective is to find values for the variables that satisfy all the equations in the system simultaneously. Systems can be presented in various forms, such as linear, quadratic, or more complex relationships. The simplest and most commonly addressed in preliminary algebra courses are linear systems, which are composed of linear equations.

A linear equation represents a straight line when graphed on a coordinate plane, and a system of linear equations represents two or more lines. The point where these lines intersect corresponds to the solution of the system, assuming one exists. When we work with systems that include two variables, the solution is a pair of numbers that, when substituted into each equation, makes both equations true.

The addition method, sometimes referred to as the method of elimination, is one technique for solving systems of equations. This method involves adding the two equations together in a way that eliminates one of the variables. To effectively use the addition method, we might first need to multiply one or both of the equations by certain factors so that, when added, one of the variables will cancel out.

This process simplifies the system into a single equation with one variable. Once we have solved for that variable, we substitute it back into one of the original equations to find the value of the other variable. It's a methodical approach that can be applied to systems of any size, but it's particularly useful for systems with two equations and two variables.

In solving systems of equations, the concept of variable elimination is crucial. This strategy is precisely what the addition method is all about—eliminating one variable at a time to eventually solve for the remaining variable(s).

Successful variable elimination hinges on manipulating the equations such that one variable cancels out when the equations are combined. This might mean multiplying each side of an equation by a number to get the coefficients of one variable to be opposites. In the example given, multiplying the second equation by -1 allowed the y terms to have coefficients that are additive inverses, (-4 and +4), leading to their elimination upon addition of the equations.

Once a variable has been eliminated, the system is easier to solve, as it's been reduced to a simpler form—a single equation with one unknown.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations look like straight lines when graphed, hence the term 'linear'. These equations are defined for two variables usually as ax + by = c, where a, b, and c are constants, and x and y represent the variables.

The solution to a linear equation is the value of the variable that makes the equation true. In the context of a system of linear equations, it's the set of values for each variable that satisfies all equations concurrently. For a system with two variables, like in the given exercise, this solution can be interpreted geometrically as the point of intersection of two lines on the Cartesian plane. If the lines intersect at one point, there is one unique solution; if they coincide, there are infinitely many solutions; and if they are parallel, there is no solution.