Problem 21 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{\frac{1}{3} x-2}=8 $$

Step-by-Step Solution

Answer

The solution is \( x = 198 \).

Step 1: Understand the Equation

The given equation is \( \sqrt{\frac{1}{3}x - 2} = 8 \). To solve this, we need to eliminate the square root by squaring both sides of the equation.

Step 2: Eliminate the Square Root

Square both sides of the equation to remove the square root: \( \left( \sqrt{\frac{1}{3}x - 2} \right)^2 = 8^2 \). This gives \( \frac{1}{3}x - 2 = 64 \).

Step 3: Isolate the Variable Term

Add 2 to both sides to eliminate the constant on the left side: \( \frac{1}{3}x - 2 + 2 = 64 + 2 \). Simplifying gives \( \frac{1}{3}x = 66 \).

Step 4: Solve for the Variable

Multiply both sides of the equation by 3 to solve for \( x \): \( 3 \times \frac{1}{3}x = 66 \times 3 \). This simplifies to \( x = 198 \).

Step 5: Verify the Solution

Plug \( x = 198 \) back into the original equation to check that it satisfies the equation: \( \sqrt{\frac{1}{3}(198) - 2} = 8 \). Simplifying inside the square root gives \( \sqrt{66 - 2} = \sqrt{64} = 8 \), confirming that \( x = 198 \) is correct.

Key Concepts

Step-by-Step Solution

Solving radical equations can seem a bit tricky at first, but breaking them down into manageable steps can simplify the process. In the given problem, our goal is to solve the equation \( \sqrt{\frac{1}{3} x - 2} = 8 \). The step-by-step solution process involves a few crucial actions:

• **Eliminating the Square Root:** Start by understanding that the square root equation \( \sqrt{...} = 8 \) requires removing the square root to solve for \( x \). The first step involves squaring both sides.
• **Isolating the Variable:** After squaring the equation, it becomes linear, making it easier to isolate \( x \) on one side of the equation.
• **Solving for \( x \):** Following algebraic rules, solve for \( x \) by performing operations like addition, subtraction, multiplication, and division as necessary.
• **Verifying the Solution:** Finally, it’s crucial to substitute the solution back into the original equation to ensure it works.

This methodical approach ensures clarity and accuracy while solving equations with radicals.

Algebraic Manipulation

Algebraic manipulation is an essential skill needed to transform equations into solvable forms. When tackling the equation \( \sqrt{\frac{1}{3} x - 2} = 8 \), algebraic manipulation plays a significant role in several stages:

• **Removing the Radical:** We started by squaring both sides of the equation to remove the radical, resulting in a simpler equation \( \frac{1}{3}x - 2 = 64 \).
• **Isolating Terms:** Next, you need to add or subtract terms to isolate the variable term on one side of the equation. In this case, we added 2 to both sides to eliminate the -2.
• **Scaling the Equation:** Finally, when dealing with fractions, it's effective to multiply both sides by the denominator to rid the equation of the fraction, as seen when we multiplied by 3 to resolve \( \frac{1}{3}x = 66 \) into \( x = 198 \).

Each of these steps involves carefully balancing both sides while maintaining equality, a core principle in algebraic manipulation.

Square Roots

Understanding square roots is foundational when solving equations involving radicals. A square root, denoted by \( \sqrt{} \), seeks to find a number which, when multiplied by itself, equals the given number under the radical. Here are some insights into square roots as they relate to solving equations like \( \sqrt{\frac{1}{3} x - 2} = 8 \):

• **Simplification by Squaring:** By squaring both sides of the square root equation, we eliminate the radical. Knowing that squaring is the inverse operation of taking the square root is key.
• **Integrity of Solutions:** After obtaining a potential solution, it’s important to confirm the outcome by substituting it back into the original square root equation. This ensures that the squared value and the radical form align correctly.
• **Square Root Values:** Familiarity with basic square roots, such as \( \sqrt{64} = 8 \), helps in verifying that operations are correctly performed and that solutions are accurate.

These concepts help dive deeper into the nature of square roots, allowing students to solve and check solutions effectively.