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The method of elimination is a strategy to solve systems of linear equations. This technique involves combining equations to eliminate one of the variables, making it possible to solve for the other variable. For instance, when working with the equations given in the exercise, the goal was to eliminate the variable y.

To execute the elimination, we looked for coefficients of y that were opposites or could be made opposites, such as -3 and 1 in this case. When one equation is multiplied or adjusted so that the coefficients of one variable are additive inverses, adding the equations cancels out that variable. Upon adding, you get a single-variable equation that's much simpler to solve. After finding the value for one variable, you can substitute it back into either of the original equations to find the value of the second variable. This step-by-step approach simplifies the system into a format that unveils the solution.

A system of linear equations consists of two or more linear equations that have the same variables and are considered simultaneously. The solution to such a system is a set of values for each variable that satisfies all equations in the system. When graphed, each equation corresponds to a line, and the intersection of these lines represents the common solution.

In the context of the exercise provided, we are working with a system of two linear equations. The objective was to find a pair of values for x and y that fits both equations. There are several ways to solve these systems, including graphing, substitution, and elimination. The elimination method, which was demonstrated in the exercise, is ideal when the coefficients of one of the variables can be easily manipulated to cancel each other out.

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions and equations to solve for unknowns. This set of techniques is fundamental in solving systems of equations by elimination. When we tackle the exercise by addition, we are employing algebraic manipulation to combine equations strategically, keeping the properties of equality in mind.

Manipulation includes operations such as adding, subtracting, multiplying, or dividing both sides of an equation by the same number to maintain balance. In the exercise, we added the two equations directly because the coefficients of y were conveniently set for elimination. However, sometimes it may be necessary to multiply the entirety of one or both equations by a number to achieve coefficients that are negatives of each other for the variable we aim to eliminate. Mastery of algebraic manipulation is not just helpful; it's crucial for efficiently and accurately solving systems of equations.