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Linear equations are fundamental components of algebra that express a relationship in which each term is either a constant or the product of a constant and a single variable. These equations can graphically be represented as straight lines on a coordinate plane, where the slope and y-intercept define the line's direction and position, respectively.

That is, a linear equation in two variables can be written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. Systems of linear equations, like the one given in the exercise, consist of two or more linear equations with the same variables. Solving such systems is about finding the values of the variables that satisfy all the equations simultaneously.

Algebraic methods involve techniques to manipulate and solve equations or systems of equations. One of these methods, used in the step by step solution, includes clearing fractions by multiplying both sides of the equation by a common denominator. Doing this simplifies the equation and makes it easier to work with.

Another foundational technique involves rearranging terms to isolate variables, such as adding or subtracting terms on both sides of an equation, or using the distributive property to expand expressions. The step by step solution began by applying this method to reformat the given equations into a more manageable form, making the subsequent steps of finding the variable values more straightforward.

The elimination method is one of several strategies to solve systems of linear equations. It involves combining equations with the goal of eliminating one of the variables. To do this, one can add, subtract, or sometimes multiply the equations by certain coefficients.

In the given solution, the elimination method is used after reformulating the equations into the standard form. By subtracting the first equation from the second, the variable \( y \) is eliminated, allowing to directly solve for \( x \). Once \( x \) is known, it is substituted back into one of the original equations to find the value of \( y \). The elimination method is favored by many due to its straightforward process and ability to effectively deal with systems that have multiple equations.

Dealing with fractions in linear equations can make the process of solving the equations more challenging. However, a solid approach is to eliminate the fractions as an initial step by finding a common denominator and multiplying all terms of the equation by it.

This approach simplifies the equation to a format without fractions, making it easier to apply further algebraic methods. As demonstrated in the exercise's solution, multiplying by the least common denominator cleared the fractions and reformatted the first equation, transforming it into a more standard form. It's crucial for students to master this fraction clearing technique as it is widely applicable and helps reduce calculation errors.