Problem 395 - Elementary Algebra

Question

In the following exercises, solve. $$ \sqrt{5 x-6}=8 $$

Step-by-Step Solution

Answer

The solution is \(x = 14\).

Step 1: Isolate the Square Root

The given equation is \(\sqrt{5x - 6} = 8\). The square root is already isolated, so move to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \(\left(\sqrt{5x - 6}\right)^2 = 8^2\). This simplifies to \(5x - 6 = 64\).

Step 3: Solve for x

Add 6 to both sides to isolate the term with \(x\): \(5x - 6 + 6 = 64 + 6\). This simplifies to \(5x = 70\). Then, divide both sides by 5: \(x = \frac{70}{5}\). Simplifying this division gives \(x = 14\).

Step 4: Verify the Solution

Substitute \(x = 14\) back into the original equation to ensure it works: \(\sqrt{5(14) - 6} = 8\). Simplifying inside the square root gives \(\sqrt{70 - 6} = \sqrt{64} = 8\), which is true.

Key Concepts

Isolating Square Roots

When solving equations that include square roots, your first task is to isolate the square root term on one side of the equation. For instance, given the equation \( \sqrt{5x - 6} = 8 \), notice how the square root is already alone. This makes our next steps easier. If the square root wasn’t isolated, you would move other terms (such as constants or coefficients) to the opposite side by adding, subtracting, multiplying, or dividing. This step keeps your equation cleaner and simpler to solve later on.

Squaring Both Sides

To remove the square root in our equation, you need to square both sides. Here's an easy way to think about it: squaring a square root cancels them out. So, if we start with \( \sqrt{5x - 6} = 8 \), squaring both sides we get \( \left( \sqrt{5x - 6} \right)^2 = 8^2 \). This simplifies to \( 5x - 6 = 64 \). Squaring both sides helps us transform the equation into a simple linear equation that we can solve using regular algebraic methods.

Solving for Variables

Once you have removed the square roots by squaring, you are left with a simpler equation. In our case, it is \( 5x - 6 = 64 \). To isolate the variable \( x \), follow these steps:

• First, add any constants to both sides. So, \( 5x - 6 + 6 = 64 + 6 \) simplifies to \( 5x = 70 \).


• Next, divide by the coefficient of \( x \). Here, divide both sides by 5 to get \( x = \frac{70}{5} = 14 \).

This systematic approach guarantees that the variable ends up isolated and you find the solution.

Verifying Solutions

Finally, always verify your solution to ensure its correctness. Substitute your found value of \( x \) back into the original equation. For \( x = 14 \), we go back to our initial equation, \( \sqrt{5x - 6} = 8 \). Plugging in \( x = 14 \), you get \( \sqrt{5(14) - 6} = \sqrt{70 - 6} = \sqrt{64} = 8 \). This confirms that our solution is correct because both sides of the original equation match. Always verify, as it confirms the correctness and ensures no mistakes were made during calculations.