Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ \sqrt{x}=7 $$

Step-by-Step Solution

Answer

The solution set is \( \{ 49 \} \).

Step 1 - Understand the Problem

We need to solve the equation \( \sqrt{x} = 7 \) for the variable \( x \). This means we are looking for a value of \( x \) that makes this equation true.

Step 2 - Isolate the Variable

To solve for \( x \), we need to eliminate the square root. Since \( \sqrt{x} = 7 \), we can square both sides of the equation to remove the square root: \( (\sqrt{x})^2 = 7^2 \). This simplifies to \( x = 49 \).

Step 3 - Check the Solution

To verify the solution, substitute \( x = 49 \) back into the original equation: \( \sqrt{49} = 7 \). Since this statement is true, our solution is correct.

Step 4 - Write the Solution Set

The solution set for the equation \( \sqrt{x} = 7 \) is \( \{ 49 \} \). This is because \( x = 49 \) is the only value that satisfies the equation.

Key Concepts

Isolating the Variable

When solving square root equations, the ultimate goal is to find the value of the variable that satisfies the equation. In our exercise, we have the equation \( \sqrt{x} = 7 \). The first step to solving this is to isolate the variable by removing the square root.

• To do this, we utilize the property of squaring. By squaring both sides of the equation, we effectively "undo" the square root.
• Therefore, squaring both sides gives us \( (\sqrt{x})^2 = 7^2 \).
• This simplification leads to the equation \( x = 49 \).

This method is foundational in algebra as it helps transform a complicated expression into a simpler form, making it easier to identify the value of the variable.

Checking Solutions

Once the variable is isolated and a potential solution is found, it's crucial to verify that the solution is correct. Checking your solution ensures that no mistakes were made during the isolating process.

• In this example, after finding \( x = 49 \) as the solution, substitute \( x \) back into the original equation.
• This means calculating \( \sqrt{49} \) and checking if it equals 7.

Since \( \sqrt{49} = 7 \) is a true statement, we've confirmed that \( x = 49 \) is indeed a correct solution. This step of verification is especially important when dealing with equations involving exponents or roots, where errors can easily occur.

Solution Set

After confirming the correctness of the solution, the next step is to write down the solution set. A solution set is a mathematical way to present all the solutions of an equation. For the equation \( \sqrt{x} = 7 \), we found only one value for \( x \) that satisfies the equation:

• The solution set is written as \( \{ 49 \} \).
• This notation signifies that 49 is the only solution.

Presenting solutions as a set is essential because it provides a clear and concise way to understand the possible values the variable can take, especially if there is more than one solution, which sometimes occurs in other types of equations.