91Ó°ÊÓ

When solving radical equations, the first step often involves removing the radical signs, which are creating complexity in the equation. To remove the square root symbols, square both sides of the equation. This action helps to simplify the equation into a form easier to work with.
For example, starting with the given exercise:
\(\text{\textbackslash sqrt{2t-7} = \text{\textbackslash sqrt{3t-12}}}\), squaring both sides, we get: \((\text{\textbackslash sqrt{2t-7}})^2 = (\text{\textbackslash sqrt{3t-12}})^2\)
This simplifies to: \(2t-7 = 3t-12\).
Now, the equation is free of radicals, and we can move on to solving for the variable.

Once the equation is free from radicals, the next step is to isolate the variable to discover its value. You should perform algebraic operations to move all terms involving the variable to one side of the equation and all constants to the opposite side:
Starting from the equation derived earlier: \(2t-7 = 3t-12\),
Subtract 2t from both sides: \[2t-7-2t = 3t-12-2t\] and simplify: \[ -7 = t-12\]
Finally, add 12 to both sides to isolate t: \[ -7 + 12 = t\] which results in: \[ 5 = t\].
Now, we have found that \( t = 5 \).
The value of the variable has been determined, but our job isn't finished yet. We need to ensure this solution is correct.

The final step is to verify whether the obtained solution satisfies the original equation. Plug the value back into the original equation to check for consistency.
Given the original equation: \(\text{\textbackslash sqrt{2t-7} = \text{\textbackslash sqrt{3t-12}}} \),
substituting \( t = 5 \), we get: \(\text{\textbackslash sqrt{2(5)-7} = \text{\textbackslash sqrt{3(5)-12}}}\), which simplifies to: \(\text{\textbackslash sqrt{10-7} = \text{\textbackslash sqrt{15-12}}}\), thus: \(\text{\textbackslash sqrt{3} = \text{\textbackslash sqrt{3}}}\).
Since both sides of the equation are equal, the solution \( t = 5 \) is verified as correct.
Verification is essential to ensure no steps were missed or incorrect during the calculation process.