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The substitution method is a cornerstone of algebra that allows us to determine whether a given number is a solution to an equation. It involves replacing the variable in the equation with the proposed number and simplifying the result to see if both sides of the equation remain equal. For instance, to check if the number 4 is a solution of the equation 5x - 1 = 3x + 2, we substitute x with 4 and obtain 5(4) - 1 on the left-hand side (LHS) and 3(4) + 2 on the right-hand side (RHS).

It's a straightforward technique yet powerful because it gives a definitive yes or no answer as to the validity of the potential solution. If after substitution, the LHS equals the RHS, the number is indeed a solution to the equation. If they do not equal, as in our case where 19 ≠ 14, the number is not a solution to the equation.

Simplifying equations is the process of performing algebraic operations to reduce an equation to its simplest form. This often involves carrying out arithmetic such as addition, subtraction, multiplication, or division to eliminate parentheses and combine like terms. When we simplify an equation, our goal is to make it as straightforward as possible to interpret or solve.

For example, when checking if 4 is a solution to the equation 5x - 1 = 3x + 2, we first multiply (5*4) to simplify the terms, getting 20. Then, we subtract 1 from 20, giving us 19 on the LHS. Similar operations on the RHS after substitution result in 14. This simplification gives us a clear picture: 19 = 14, which, after simplification, reveals that the two sides are not equal, thus the number 4 is not a solution to the original equation.

Linear equations form the basis for much of algebra and are equations of the first degree, meaning they have variables raised only to the power of one. These equations can have one or more variables but are most commonly seen with just one, as in the equation 5x - 1 = 3x + 2. The graph of a linear equation is a straight line, hence the name 'linear'.

Such equations are straightforward to solve using a variety of methods, including graphing, substitution, and elimination. In our case, by attempting the substitution method, we concluded that the number 4 does not satisfy the equation as the two sides do not equate post-simplification. This is part of the beauty of linear equations: they are simple to manage, and methods like substitution allow for quick solutions and verifications.