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Solution verification is a crucial step when dealing with linear equations. It involves checking whether a proposed solution satisfies the equation. This means when you substitute the solution into the equation, the left-hand side (LHS) should equal the right-hand side (RHS).

For our exercise, the goal was to check if the number \(4\) is a solution for the equation \(12 - 2a = 18\). When the student substituted \(4\) for \(a\), they ended up with \(12 - 8 = 4\).

For a valid solution, this result would need to match the RHS of the original equation, which is \(18\). Since \(4 eq 18\), we verify that \(4\) is not a solution. It's very important to remember that solution verification helps us ensure that our results are accurate. This can prevent errors later on in more complicated problems.

The substitution method is a technique used to solve equations by replacing variables with their respective values. Here, it allows us to determine if a given number is a solution. By taking the provided number and substituting it into the equation, you convert it into a simpler arithmetic problem.

In this example, after substituting \(4\) for \(a\) in the equation \(12 - 2a = 18\), the expression becomes \(12 - 2(4) = 18\). This step transforms the problem from a puzzle of an unknown variable into a straightforward calculation, making it easier to work through the rest of the process.

Using substitution helps to simplify the problem right away. It reduces the potential for mistakes, especially when handling more complex equations where solving directly could be tricky.

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve or verify. In this context, after substituting our number (4) into the equation, we perform some arithmetic operations. These operations are part of the algebraic manipulation.

Starting with the expression \(12 - 2(4)\), you first multiply the values in the parentheses: \(2 \times 4 = 8\). Then, you carry out the subtraction: \(12 - 8\). This gives you the result \(4\).

These steps of simplifying and reducing are what we call algebraic manipulation. It's a process of breaking down complex expressions into more understandable parts. Practicing these skills can help in tackling a wide variety of algebra problems, making sure you can always find a path to the solution.