91Ó°ÊÓ

To solve radical equations, the first step is usually to eliminate the square roots. This makes the equation easier to handle. In the given equation \(\sqrt{3t + 4} = \sqrt{4t + 3}\), we see square roots on both sides. To remove them, you need to square both sides of the equation. When you square a square root, it cancels the square root, leaving you with just the inside expression. Here, squaring both sides gives: \[ (\sqrt{3t + 4})^2 = (\sqrt{4t + 3})^2 \] This simplifies to \[ 3t + 4 = 4t + 3 \] By removing the square roots, we transformed our equation into a simpler linear equation. Remember to be careful when squaring both sides because this process can sometimes introduce extraneous solutions, which we'll discuss in a later section.

After eliminating the square roots, the next step is to isolate the variable. In our simplified equation \[ 3t + 4 = 4t + 3 \], our goal is to solve for the variable \(t\). Start by getting all the terms containing \(t\) on one side and constant terms on the other. We can do this by subtracting \(3t\) from both sides of the equation: \[ 3t + 4 - 3t = 4t + 3 - 3t \] This simplifies to \[ 4 = t + 3 \] Now, we need to isolate \(t\). We do this by subtracting 3 from both sides: \[ 4 - 3 = t + 3 - 3 \] \[ 1 = t \] Hence, our isolated variable is \(t = 1\). This isolation process is fundamental in algebra, as it allows you to solve for the required variable by getting rid of other numbers on that side of the equation.

Once you've solved the equation and found your variable, it's essential to verify the solution. This step checks if the solution satisfies the original equation. Start by substituting the found variable back into the original equation. For example, our solution was \(t = 1\). Plugging it back into the original equation \ (\