Problem 25 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{6 x-2}-3=7$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 17 \).

Step 1: Isolate the square root

First of all, isolate the square root expression. To do this, add 3 to both sides of the equation: \[ \sqrt{6 x-2} = 7 + 3 \rightarrow \sqrt{6 x-2} = 10 \]

Step 2: Eliminate the square root

To eliminate the square root, square both sides of the equation: \[ (\sqrt{6 x-2})^2 = 10^2 \rightarrow 6x - 2 = 100 \]

Step 3: Solve for x

Now solve the remaining equation for x. Add 2 to both sides and then divide by 6: \[ 6x = 100 + 2 \rightarrow 6x = 102 \rightarrow x = \frac{102}{6} \rightarrow x = 17 \]

Step 4: Check for extraneous solutions

Substitute the possible solution back into the original equation to check if it's valid: \( \sqrt{6 * 17 -2}-3 = 7 \). Simplifying gives \( \sqrt{100}-3 = 7 \) and further simplification gives \( 10 - 3 = 7 \). Since the left side equals the right side, x = 17 is indeed a solution to the original equation.

Key Concepts

Extraneous Solutions

In solving algebraic equations, especially those involving square roots, we encounter something known as extraneous solutions. These are results that emerge from the process of solving the equation but are not true solutions to the original equation. They typically arise when we perform operations that are not reversible, such as squaring both sides of an equation, which can introduce false solutions.

To identify extraneous solutions, it's crucial to substitute the found values back into the original equation. If the equation holds true with the substituted values, then the solution is valid; otherwise, it is extraneous. For instance, if we mistakenly obtained a solution that results in the square root of a negative number (which is not possible in the set of real numbers), it would be considered extraneous. This checking process ensures the reliability of our solutions and is a fundamental step in algebraic problem-solving.

Isolating the Square Root

When faced with square root equations, one of the initial and most important steps is to isolate the square root. This means rearranging the equation so that the square root term stands alone on one side of the equality. This typically involves using basic algebraic operations such as addition or subtraction.

For example, consider the equation \( \sqrt{6x - 2} - 3 = 7 \). To isolate the square root, you would add 3 to both sides, yielding \( \sqrt{6x - 2} = 10 \). This clear separation between the square root and the rest of the terms enables the subsequent steps, like squaring both sides, to eliminate the square root and further solve for the variable. Isolating the square root simplifies the equation and sets the stage for solving it without complications.

Checking Solutions in Algebra

The cornerstone of solving equations in algebra, particularly after manipulating them, is checking solutions. After finding potential solutions, it's important to verify their correctness by substituting them back into the original equation. This step is essential to confirm that the operations performed did not introduce any incorrect solutions and that the solution satisfies all components of the equation.

For example, after solving \( \sqrt{6x - 2} - 3 = 7 \) and finding \( x = 17 \), we must check it. Plugging 17 back into the original equation, we get \( \sqrt{6 \times 17 - 2} - 3 = 7 \), which simplifies to \( \sqrt{100} - 3 = 7 \) and ultimately to \( 10 - 3 = 7 \), confirming that the solution is correct. This verification step ensures that our solution is not extraneous and that it fully complies with the given equation.