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The method of substitution is a fundamental algebraic technique used to solve systems of linear equations. It involves replacing one variable with an equivalent expression obtained from another equation within the system. This technique is particularly useful when one equation can be easily solved for one variable in terms of the others.

For instance, with our given system of equations, the second equation is chosen to isolate variable 'x'. Once 'x' is expressed in terms of 'y', this expression is substituted into the first equation in place of 'x'. This reduces the system from two equations in two variables to a single equation in one variable which is then much easier to solve. Using substitution avoids the need to manipulate both equations simultaneously, thus simplifying the problem-solving process.

Solving systems of linear equations is essential for finding the values of unknowns that satisfy all equations in the system simultaneously. Put another way, we are looking for the point(s) of intersection between lines, planes, or hyperplanes represented by these equations in n-dimensional space.

To solve such a system, we often leverage methods like graphing, substitution, elimination, and matrix operations. The substitution method, in particular, is advantageous when the system includes an equation that can be conveniently solved for one variable. Through step-by-step replacement and simplification, the system's complexity is broken down until all variable values are determined, yielding a solution that satisfies all the original conditions.

Isolation of variables involves manipulating an equation to have the subject variable on one side of the equation, set apart from the other variables or constants. This is especially helpful when preparing to use the substitution method for solving systems of equations.

In our exercise, the first step is critical: choosing the most suitable equation to isolate a variable. By isolating 'x' in the second equation, we set up a straightforward pathway to substitute this expression into other equations of the system. This technique enables a clear and sequential path to solution, as it minimizes the potential for algebraic error and clarifies the problem-solving process. The goal is to make the equation look like 'variable = expression', ensuring a smooth substitution.

Algebraic substitution is the act of replacing variables with their equivalent values or expressions. After isolating a variable in one equation, we substitute that expression into another equation—exactly what happens within the substitution method.

Returning to our example, after isolating 'x', we substitute the expression for 'x' into the first equation. This reshapes the equation into a single variable format, in this case, 'y' only, which can then be solved directly. Algebraic substitution is not only used in solving systems of equations but also in simplifying expressions, integrating functions, and many other contexts within mathematics. It's a versatile tool that reduces problems to a more manageable size by methodically eliminating variables one by one.