Problem 59 - Algebra 1

Question

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{6 x-2}-3=7$$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{6x - 2}-3 = 7 \) is \( x = 17 \)

Step 1: Simplify the Equation

First of all, simplify the equation in order to isolate the square root on one side. Given equation is \( \sqrt{6x - 2} - 3 = 7 \). Adding 3 to both sides gives \( \sqrt{6x - 2} = 10 \).

Step 2: Square Both Sides

To remove the square root, square both sides of the equation. This gives \( (6x - 2) = 10^2 \) which simplifies to \( 6x - 2 = 100 \).

Step 3: Solve for x

Rearrange the equation to solve for x. Add 2 to both sides, giving \( 6x = 102 \). Then divide both sides by 6 to obtain the solution \( x = \frac{102}{6}=17 \).

Step 4: Graphing

Graph the function \( y=\sqrt{6x - 2}-3 \) and the line \( y=7 \) using a graphing calculator. The x-coordinate of the intersection point will be the solution to the equation.

Step 5: Checking Algebraically

To ensure the solution is not extraneous, substitute x=17 back into the original equation: \( \sqrt{6*17 - 2}-3 \)= 7. Evaluate this to verify that the solution is valid.

Key Concepts

Graphing Calculator Usage

Using a graphing calculator is an efficient way to solve and visualize radical equations. For instance, when given the equation \(\sqrt{6x-2}-3=7\), a graphing calculator helps to provide a visual representation of the problem. To use one effectively, start by entering the radical function \(y=\sqrt{6x - 2}-3\) into the calculator. Then, plot the constant function \(y=7\). The intersection point(s) between the two graphs represent the solution(s) to the equation. This graphical method not only gives you the answer but also enables you to understand the behavior of the radical equation graphically.

Remember, graphical solutions may not always be exact due to pixel limitations on the screen. Therefore, after finding an approximate solution with the graphing calculator, it's important to check the solution algebraically to ensure precision.

Isolate Square Root

Isolating the square root in a radical equation is a crucial step before you can solve for the variable. In the given equation \(\sqrt{6x-2}-3=7\), the goal is to have the square root term, \(\sqrt{6x-2}\), by itself on one side of the equation. You achieve this by performing inverse operations that 'undo' what is being done to the square root. As shown in the solution steps, you add 3 to both sides to get \(\sqrt{6x-2}=10\). Isolating the square root prepares you to square both sides in the next step and is essential for simplifying radical equations.

Square Both Sides

To solve for the variable within a square root, you square both sides of the equation. This is because squaring a square root will eliminate the radical sign, leaving you with a more familiar linear or quadratic equation. In the example \(\sqrt{6x-2} = 10\), squaring both sides results in \(6x-2 = 10^2\), which simplifies further to \(6x-2 = 100\). Squaring both sides is a powerful tool that helps you progress towards finding the variable; however, this action can introduce extraneous solutions, so it is imperative to check your answer algebraically after you've found a potential solution.

Check Solutions Algebraically

Although graphing calculators aid in visualizing solutions to radical equations, it is necessary to check solutions algebraically to ensure they are not extraneous. After solving the equation and finding that \(x = 17\), it's crucial to substitute this value back into the original radical equation to verify its accuracy. By substituting \(x\) with 17, the equation \(\sqrt{6*17 - 2}-3\) should simplify to 7. If both sides of the original equation are equal after the substitution, the solution is valid. If they are not, the solution is extraneous, and one would need to re-examine the earlier steps for possible errors. This process vindicates the accuracy of your solution and is the final step to ensure your answer is not only possible but certainly correct.