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When dealing with radical equations, you are solving for the unknown variable within the radicand—the expression under the square root. These equations can present a unique challenge since the radical can affect the solutions in unexpected ways.

Great care should be taken when solving radical equations, since the process of squaring both sides can introduce extraneous solutions—solutions that fit the modified equation but not the original. That's why after finding potential solutions, it's important to plug them back into the original equation and verify their validity.

For instance, our exercise presented the equation \( \sqrt{y+7} = 9 \). Here, we initially have a single root, but this does not inherently guarantee a single solution. Always remember, possible additional or extraneous solutions need to be checked in the original equation for their true validity.

The process of isolating variables is a foundational technique in algebra and serves as a crucial stepping stone in solving equations. To isolate the variable means to get the variable alone on one side of the equation, often by performing inverse operations.

For example, if the variable is added to a number, subtract the same number from both sides. If it is multiplied, divide both sides by the number, and so forth. This systematic approach grants clarity and simplifies the equation, as seen in the latter part of our exercise, where subtracting 7 from both sides of \($y + 7 = 81\) yielded a clean solution for \(y\).

It's vital to do the same to both sides of the equation to maintain its balance—a central tenet of algebra. Remember that solving an equation is like keeping scales balanced; what you do to one side, you must do to the other.

The technique of squaring both sides of an equation is often employed to remove square roots. When you square a square root, you're left with the radicand; squaring the other side, however, must be done with precision to avoid errors.

As shown in step 1 of the solution, when we squared \(\sqrt{y+7}\) and the number 9, we were left with \(y + 7 = 81\). However, this step makes the assumption that only the positive square root was considered. Remember that squaring a number yields a positive result, whether the original number was positive or negative. This property doesn't typically affect equations with single square roots, like our example, but is essential to remember in more complex equations that involve multiple terms and roots.

When applying this technique, it's paramount to thoroughly check the solutions in the initial equation, as it’s possible for the squaring process to introduce 'solutions' that don't actually fit the original equation.