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A square root equation involves finding the square root of a particular expression. In the example provided, we have an equation given by \( \sqrt{2y - 1} = 9 \). The goal is to determine the value of \( y \) that satisfies this equation. The presence of the square root implies that raising both sides to the power of 2 will eliminate the root. By squaring both sides, the equation simplifies from the form \( \sqrt{2y - 1} \) to \( 2y - 1 = 81 \).

The process of eliminating the square root is crucial, as it transforms the problem into a more straightforward algebraic equation that is easier to solve. Keep in mind that squaring changes the equation's structure, making it vital to verify the solution later to avoid extraneous results.

Verification is a crucial step in solving equations, especially when dealing with operations like squaring both sides, which may introduce extraneous solutions. An extraneous solution is a solution that emerges from the process of solving the equation but doesn't satisfy the original problem. To confirm the correctness, we substitute the solution back into the original equation.

In our example, substituting \( y = 41 \) back into the original equation gives us:

• Replace \( y \) with \( 41 \): \( \sqrt{2(41) - 1} \)
• Calculate inside the square root: \( \sqrt{82 - 1} = \sqrt{81} \)
• The value of \( \sqrt{81} \) indeed equals \( 9 \)

As the left side equals the right side, it confirms that \( y = 41 \) is a valid solution. Verification serves as a double-checking tool, ensuring the result is correct and applicable.

Algebraic manipulation is the technique of rearranging and simplifying equations to find the solution for the unknown variable. In our given equation, after squaring both sides to eliminate the square root, the task becomes straightforward algebraic manipulation.

First, you have \( 2y - 1 = 81 \). Here's how you solve it step by step:

• Add 1 to both sides of the equation to avoid subtraction: \( 2y = 82 \)
• Divide both sides by 2 to isolate and solve for \( y: y = 41 \)

Each manipulation step changes the equation closer to finding the value of \( y \). These steps might seem basic but mastering them aids in solving more complex problems. Breaking it down like this ensures clarity and accuracy in finding solutions.