Solving algebra word problems involves systematic problem-solving steps. Let's break it down:
- 1. **Express the given relationship:** Start by writing out the relationship between the given quantities using an equation. In our example, that is: \[ 2c = \sqrt{\frac{a}{b}} \].
- 2. **Isolate the target variable:** Manipulate the equation to solve for the desired variable. Here, we isolated b: \[ 4c^2 = \frac{a}{b} \]and then expressed b as: \[ b = \frac{a}{4c^2} \].
- 3. **Substitute given values:** Use the provided information to find the value of the variable. With given values \(a = 8\) and \(c = 2\), we substituted them into our equation: \[ b = \frac{8}{4(2)^2} = \frac{8}{16} = \frac{1}{2} \].
- 4. **Verify with additional data:** Use any other provided data to check your results. In our solution, the verification step was crucial. We were given: \[ \frac{c^2}{a} = \frac{1}{2} \].By substituting \(a = 8\) and \(c = 2\): \[ \frac{2^2}{8} = \frac{4}{8} = \frac{1}{2} \].The condition is satisfied, confirming our solution is correct.
Breaking down problems into these steps ensures clarity and accuracy, making it easier to solve complex algebraic problems.