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When dealing with algebraic equations, we're referring to mathematical statements that show the equality between two expressions. In the context of our problem, the equation \(4z = 20\) shows a relationship involving the variable \(z\).
Algebraic equations often contain variables which are the unknowns we need to solve for. Variables give equations flexibility and are typically represented by letters like \(z\) or \(x\). These equations are integral to understanding various real-world problems, as they help us describe relationships and predict outcomes.
To solve an algebraic equation, the goal is to find the value of the variable that makes the equation true. The process involves isolating the variable on one side of the equation to find what it equals. Mastering such concepts enabled us to solve more complex problems and understand real-world scenarios better.

The substitution method is a straightforward technique utilized to determine whether a proposed value is the solution to an equation. This involves replacing the variable in the equation with the given number to check for validity. Let's dive into how this works!
Consider we are tasked with determining if \(10\) is a solution for the equation \(4z = 20\). The substitution method involves the following:

• Replace the variable \(z\) with \(10\), giving us \(4 \times 10\).
• Calculate the result of this substitution, which would be \(40\).

This step confirms if the left side of the equation equals the right side when a specific number replaces the variable. If both sides match, the number is a solution. If not, as in our scenario where \(40\) does not equal \(20\), the number isn't a solution.
Substitution is a fundamental method in algebra because it provides a systematic way to test potential solutions, ensuring accuracy when evaluating complex equations.

Solution verification is a crucial step in solving equations to ensure the accuracy of your solution. This involves checking whether the substituted value truly satisfies the given equation.
The verification process entails a direct comparison of the final computation after substitution with the equation's right side to confirm correctness. In our example:

• Compute: Substitute \(10\) into \(4z = 20\) which transforms it into \(40 = 20\).
• Analyze: Check if the left-hand side (\(40\)) equals the right-hand side (\(20\)).

If the two sides of the equation are equal, the chosen number is confirmed to be a solution.
Verification is not limited to just correctness; it's also about gaining confidence in your approach and methods. This practice helps catch errors early, improving problem-solving skills and mathematical intuition.