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Algebraic equations form the foundation of algebra and represent relationships between variables and numbers. An equation is comprised of two expressions set equal to each other, connected by an equal sign. It usually contains one or more unknown values, commonly represented by letters, like the 'y' in the equation from the exercise, \(30 - y = 10\).

The process of solving an equation involves finding the value of the variable that makes the equation true. In other words, you're looking for the exact number that, when substituted in place of the variable, will turn the expression on one side of the equal sign into the exact same value as the expression on the other side.

Understanding how to manipulate these equations is pivotal for solving various mathematical and real-world problems.

The substitution method is a technique employed to solve algebraic equations by replacing variables with their respective values. To visualize this, consider an empty box representing the variable. If we know what should be inside the box, we replace the empty box with the actual content. Similarly, when given a particular value for a variable, we substitute it directly into the equation in place of the variable.

For Example:
Let's operate on our original equation \(30 - y = 10\) and replace the \(y\) with 20, obtaining \(30 - 20\). This simplifies the process and shows us whether the chosen value for \(y\) satisfies the equation, leading us to an equality. Using substitution makes the complex task of solving for variables a much more straightforward and organized procedure.

Verifying the solution of an equation is a crucial final step to ensure accuracy in the problem-solving process. Just because an answer appears correct does not mean the problem is solved; it needs to be validated. When you plug the solution back into the original equation and simplify, both sides of the equation must be equal.

Let's apply this to our exercise. After substituting \(y=20\) into \(30 - y = 10\), we get \(30 - 20 = 10\). The left side simplifies to 10, which is indeed equal to the right side. Thus, the equation holds true, confirming that \(y=20\) is a valid solution. This step – called validation – seals the deal and provides confidence in the answer. Leaving out validation could mean missing unnoticed errors, so it's always a good practice to include this step and ensure the values you determined genuinely satisfy the equation.