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Understanding how to solve systems of equations is foundational in algebra. A system of equations consists of two or more equations with the same set of variables. The goal is to find a set of values that satisfy all equations in the system simultaneously. There are several methods to solve these systems, including graphing the equations, using substitution, the addition (or elimination) method, and matrix operations.

Here, we focus on the addition method, which involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. The addition method is particularly useful when the coefficients of one of the variables are opposites because adding the equations directly eliminates that variable.

The algebraic method refers to solving systems of equations using algebraic operations to manipulate the equations. Unlike graphical solutions, which involve sketching lines or curves on a plane, the algebraic approach is precise and follows a set of steps that lead to the exact solution.

For instance, in the given exercise, the addition method is an algebraic technique where equations are combined to eliminate a variable. To do this effectively, it's essential to manipulate the equations if they're not already set up for easy elimination. Beyond addition, you may need to multiply or divide terms to align the variables for elimination. This method is foundational in algebra and is applied across various fields of mathematics and science.

After finding the values for the variables in a system of equations, it's critical to verify that the solutions are correct. Verification ensures your solution actually satisfies each original equation. To verify, plug the values back into each given equation and check if the left side equals the right side.

In the provided exercise, after determining that the proposed solution is \(x=2, y=-1\), you would replace \(x\) and \(y\) in each initial equation. If both equations balance with these values, then you can confidently claim that the solution is correct. This step, while it may seem trivial, is crucial as it validates the hard work done in the problem-solving process.

While the addition method works efficiently for the given exercise, another standard method for solving systems of equations is the substitution method. This approach involves solving one of the equations for one variable in terms of the other and then substituting this expression into the other equation. It's a step-by-step process where the ultimate goal is to isolate one variable and solve for it.

Taking the example \(x+y=1\), if you solved for \(y\), you would get \(y=1-x\). You would then substitute \(1-x\) for \(y\) in the other equation \(x-y=3\), leading to \(x-(1-x)=3\). Although the substitution method wasn't necessary for the given exercise, it's essential to be versatile and familiar with different techniques to solve various systems of equations you may encounter.