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The addition method, also known as the elimination method, is a technique used to solve systems of linear equations. It involves combining the two equations in such a way that one of the variables is eliminated. This simplifies solving for the remaining variable. For our given system: \(-x + 2y = 4\) and \(x - 5y = 1\), we add both equations to eliminate the \(x\) variable. By doing so, we focus on solving for \(y\). Once \(y\) is found, it can be substituted back into one of the original equations to find \(x\). This method is particularly useful when the coefficients of one variable are opposites, making elimination straightforward.

The elimination method is another name for the addition method. It is used to solve systems of linear equations by eliminating one of the variables through addition or subtraction. In our example, we eliminated \(x\) by adding the two equations: \(-x + 2y + x - 5y\). This results in: \(-3y = 5\). By solving for \(y\), we find \(y = \frac{-5}{3}\). Next, we substitute \(y\) back into one of the original equations to solve for \(x\). This method often requires manipulating the equations through multiplication or division to align the coefficients for easy elimination. Here, we didn't need such adjustments as the \(x\) coefficients were already opposites.

Linear equations are mathematical statements that describe a straight-line relationship between variables. They can be expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The given system: \(-x + 2y = 4\) and \(x - 5y = 1\) consists of two linear equations. Solving such systems involves finding the values of \(x\) and \(y\) that satisfy both equations simultaneously. Techniques like the addition method help solve these systems by simplifying the equations and making the unknowns more accessible. In real-world applications, linear equations are used in various fields such as physics, engineering, and economics to model relationships and predict outcomes.