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The substitution method is a powerful tool for solving algebraic equations, especially when we are given a specific value for a variable and we need to determine whether it is a solution. In the exercise, we have the equation \(4z = 20\) and we are asked if \(10\) can be the mysterious \(z\) we are looking for. To use substitution, we replace every instance of \(z\) with \(10\). After the replacement, we get \(4 \times 10\), which simplifies to \(40\). This is not equal to \(20\), thus revealing that \(10\) is not the correct value for \(z\). Substitution is like trying on a shoe to see if it fits; if it does, we have a solution, but if it doesn’t, we keep looking.

When applying the substitution method, it's important to carefully substitute the given value into all occurrences of the variable and to follow the order of operations when simplifying the expression. As you practice, the process becomes second nature, helping you verify solutions quickly and accurately.

Variable evaluation is like being a detective, where every equation is a case and the variables are clues. In our investigation with the equation \(4z = 20\), variable evaluation means taking the given clue—in this case, the possible solution \(10\)—and scrutinizing it closely. We evaluate the variable by replacing it in the equation and performing the necessary calculations.

By substituting \(10\) for \(z\), we perform the multiplication \(4 \times 10\) to get \(40\). Just like a detective deduces facts, we deduce that since \(40\) does not equal \(20\), our suspect, \(10\), cannot be the solution. The skill of evaluating variables helps us not only determine right solutions but also to eliminate wrong ones, making our overall mathematical detective work much more efficient.

Verifying solutions of equations is a crucial step in solving algebraic problems. It's similar to a judge ruling if the case before them stands or falls. After hypothesizing that a number is a solution, we must prove it. For \(4z = 20\), we hypothesized that \(10\) was the solution and verified it through the substitution method.

The verification process involved substituting \(10\) into the equation and calculating the result. Since \(40 eq 20\), our initial hypothesis was incorrect. Verifying solutions guarantees that our answers make sense in the given mathematical context. Sometimes, a single equation may even have multiple solutions, and other times none at all. Regularly verifying solutions helps avoid mistakes and builds solid mathematical foundations where guesses are transformed into certainties.