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When solving algebraic equations, the substitution method is a cornerstone technique that allows students to determine if a particular value is a solution to the equation. This involves replacing the variable in the equation with the given number and simplifying the result to see if the two sides of the equation are equal.

The beauty of this method lies in its simplicity - if after substitution the equation holds true, then the number used is indeed a solution. If not, as seen in our original exercise, the number does not satisfy the equation. The substitution method not only provides a means of verification but also enhances understanding of the relationship between variables and constants within equations.

Algebraic expressions are the vocabularies of algebra and constitute a mix of numbers, variables, and operation symbols that represent a specific quantity. They can be as simple as a single number, 'x', or as intricate as a polynomial. In the context of solving equations, understanding how to manipulate these expressions is crucial.

Each operation performed on an expression, whether it be addition, subtraction, multiplication, or division, follows well-defined rules such as the distributive property or combining like terms. Mastering these rules allows students to successfully simplify or expand expressions, which is an essential step in solving equations and accurately conducting the substitution method.

Equation solutions are the values which, when substituted into an equation, satisfy it, meaning both sides of the equation are equal. An equation may have a single solution, multiple solutions, or in some cases, no solution at all. The process of finding these solutions often involves isolating the variable, or expressing it in its simplest form relative to the other numbers and terms present.

The exercise showcased how -10, when substituted into the given equation, did not satisfy it as it did not yield an equal balance of the two sides. This shows the importance of checking your work when dealing with algebra - even when equations appear straightforward. The goal is ultimately to understand when and why certain numbers qualify as solutions, which lays the groundwork for more complex problem-solving down the line.