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When you face a pair of equations with two variables, such as \[3x - 7y = 13\] and \[6x + 5y = 7,\] you're dealing with a linear system. Solving linear systems involves finding the set of values for the variables that make both equations true simultaneously. There are various methods to solve such systems, and one popular technique is the addition method, also known as the elimination method.

The goal of the addition method is to eliminate one variable so that you can solve for the other. To do so, you often need to manipulate the equations to get the coefficients of one variable to be the same or opposites. Then, by adding or subtracting the equations from each other, one variable can be eliminated. In our example, the y variable was eliminated after manipulating and adding the equations, leading to a single-variable equation that was simple to solve.

Algebraic manipulation is the process of rearranging and transforming equations using algebraic rules. This might mean multiplying, dividing, adding or subtracting terms on both sides of an equation to achieve the desired form. For instance, in the step-by-step solution we provided, we multiplied the entire first equation by 5 and the second equation by 7. The rationale behind this specific alteration was to make the coefficients of y equal and opposite, hence allowing us to later add the equations effectively eliminating the y variable.

Algebraic manipulation is a foundational skill in solving mathematical problems and applies to a wide variety of problems beyond linear equations. Understanding how to carefully perform these operations while maintaining the equality of the equations is crucial for obtaining accurate solutions.

The substitution method is another strategy used to solve systems of linear equations, where one equation is solved for one variable and then substituted into the other equation. This makes the system easier to manage, as it reduces to a single equation with one variable.

In the given exercise, we used a combination of the addition and substitution methods. After using the addition method to find the value of x, we then substituted this value back into one of the original equations. This process of back-substituting allowed us to find the value of y.

Substitution is particularly handy when one of the equations is already solved for one of the variables or can be easily manipulated to do so. It is also critical to pick an equation for substitution that makes the calculation as simple as possible, usually, the one that involves fewer steps to solve.