Problem 25 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(5 \sqrt{t-1}=6\)

Step-by-Step Solution

Answer

The solution is \(t = \frac{61}{25}\).

Step 1: Isolate the Square Root

Given the equation \(5 \sqrt{t-1}=6\), divide both sides by 5 to isolate the square root term: \[\sqrt{t-1} = \frac{6}{5}.\]

Step 2: Eliminate the Square Root

Square both sides of the equation to eliminate the square root:\[(\sqrt{t-1})^2 = \left(\frac{6}{5}\right)^2.\]This simplifies to:\[t-1 = \frac{36}{25}.\]

Step 3: Solve for t

Add 1 to both sides of the equation to solve for \(t\):\[t = \frac{36}{25} + 1.\]Convert 1 to a fraction with a denominator of 25: \[t = \frac{36}{25} + \frac{25}{25} = \frac{61}{25}.\]

Step 4: Verify the Solution

Substitute \(t = \frac{61}{25}\) back into the original equation to verify:\[5 \sqrt{t-1} = 5 \sqrt{\frac{61}{25} - 1} = 5 \sqrt{\frac{61}{25} - \frac{25}{25}} = 5 \sqrt{\frac{36}{25}} = 5 \times \frac{6}{5} = 6.\]The left-hand side equals the right-hand side, confirming the solution is correct.

Key Concepts

Isolating the Square Root

To solve an equation involving a square root, the first step is to isolate the square root term on one side of the equation. This means that we want the square root by itself, without any other numbers or terms multiplied by it. In the example given, the equation is \(5 \sqrt{t-1} = 6\). To isolate the square root, divide both sides of the equation by 5, which gives us:

• \(\frac{5 \sqrt{t-1}}{5} = \frac{6}{5}\)
• \(\sqrt{t-1} = \frac{6}{5}\)

This crucial step simplifies the problem significantly and sets us up for the next step. Isolating the square root makes it easier to manage the equation and helps in removing the square root, making the equation solvable in simpler terms.

Eliminating the Square Root

Once the square root is isolated, the next step is to eliminate it. This is done by squaring both sides of the equation. Remember, squaring a square root cancels out the square root itself. Based on the isolated equation \(\sqrt{t-1} = \frac{6}{5}\), we square both sides:

• \((\sqrt{t-1})^2 = (\frac{6}{5})^2\)
• \(t - 1 = \frac{36}{25}\)

Squaring removes the square root, leaving us with a simpler equation to solve: \(t - 1 = \frac{36}{25}\). This step is fundamental to transforming the initial equation into a basic linear equation.

Verifying the Solution

Once we have a potential solution, it is important to verify that it satisfies the original equation. In our example, after solving the equation, we found that \(t = \frac{61}{25}\). To verify, substitute \(t = \frac{61}{25}\) back into the original equation:

• \(5 \sqrt{t-1} = 5 \sqrt{\frac{61}{25} - 1}\)
• This simplifies to \(5 \sqrt{\frac{36}{25}}\)
• Which further simplifies to \(5 \times \frac{6}{5} = 6\)

Since the left side of the equation equals the right side, our solution is verified. Verification ensures that the solution is correct and was not affected by any potential extraneous solutions introduced during the squaring process.