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Chapter 3: Problem 13

Solve each equation. If it has no solution, write "no solution." $$ \sqrt{\frac{x}{7}}=3 $$

Short Answer

Expert verified
x = 63.

Step by step solution

01

Isolate the Square Root

The given equation is \(\frac{\text{x}}{7}^{0.5}=3\). Notice that the square root \(\frac{\text{x}}{7}^{0.5}\) is already isolated. This is the simplest form to start solving the equation.
02

Square Both Sides

Eliminate the square root by squaring both sides of the equation. \[ \bigg( \frac{\text{x}}{7}^{0.5} \bigg)^2 = 3^2 \] Simplifying this, we get: \[ \frac{x}{7} = 9 \]
03

Solve for x

To solve for \(\text{x}\), multiply both sides of the equation by 7 to isolate \(\text{x}\). \[ \frac{\text{x}}{7} \times 7 = 9 \times 7 \] This yields: \[ \text{x} = 63 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Square Root
When solving radical equations, the very first step is to isolate the square root on one side of the equation. This makes it easier to eliminate the square root later on. For this specific problem, we start with the equation \(\sqrt{\frac{x}{7}}=3\). Here, the square root, \(\sqrt{\frac{x}{7}}\), is already isolated on the left side. Having the square root isolated, we can move on to the next step without any additional changes to the equation.
Squaring Both Sides
The second step involves removing the square root by squaring both sides of the equation. By doing this, we effectively 'undo' the square root operation. Given our equation \(\sqrt{\frac{x}{7}}=3\), squaring both sides leads to:
\[ \left( \sqrt{\frac{x}{7}} \right)^2 = 3^2 \]
The square and square root cancel each other out on the left side, simplifying to:
\[ \frac{x}{7} = 9 \]
Now, the equation no longer has a radical, making it much simpler to solve.
Equation Simplification
Now that we've eliminated the square root, we can simplify the equation to solve for \(x\). Starting from the equation
\(\frac{x}{7} = 9\),
we need to isolate \(x\). To do this, multiply both sides of the equation by 7:
\[ \frac{x}{7} \times 7 = 9 \times 7 \]
Simplifying the right side, we get:
\[ x = 63 \]
Therefore, the solution to the equation \(\sqrt{\frac{x}{7}}=3\) is \(x = 63\). Always double-check your solution by substituting it back into the original equation to ensure it satisfies the equation.

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