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Algebraic methods are pivotal techniques in solving mathematical problems, especially equations. They include strategies such as substitution, elimination, and graphical methods to find the solutions to systems of equations. These systems can often represent real-world problems, where variables are unknown quantities we seek to determine.

Consider the process as if you are translating a complex story into a set of simpler, mathematical statements. The craft of algebra involves manipulating these statements (equations) in various ways to isolate variables and determine their values. This process often includes combining like terms, using the distributive property, and familiar operations such as addition, subtraction, multiplication, and division to simplify equations and ultimately reach a solution.

The Substitution method is a cornerstone algebraic technique used to solve systems of linear equations, where one equation is solved for one variable and that expression is substituted into another equation. This method is particularly useful when equations are already solved for a specific variable, as in the case of our given exercise with two equations solved for v.

The beauty of this method lies in its simplicity. You perform a straight swap: wherever you see the variable in one equation, you replace it with its equivalent expression from the other equation. It's akin to piecing together parts of a puzzle; you are trying to find the piece that fits perfectly to complete the picture of the solution. The result is a single equation with one variable that can be solved using basic algebraic operations.

Simultaneous equations are a set of equations with multiple variables that are true at the same time. The 'simultaneous' part means that each equation swings in harmony with the other; the solution must satisfy all the equations in the system simultaneously. In the context of our example, we have two equations that describe a consistent relationship between v and t.

With simultaneous equations, you can imagine being a detective, finding clues (equations) that must all point to the same solution (values of variables). The true art in handling simultaneous equations is not just about finding the values that work for one equation, but the pair of values that resolve the entire system, a skill that proves invaluable not just in pure mathematics, but in everyday problem-solving scenarios as well.