Problem 46 - Algebra 2 and Trigonometry

Question

Solve and check each equation. \(7-b \sqrt{8}=b \sqrt{5}+4\)

Step-by-Step Solution

Answer

\(b = \frac{3}{\sqrt{8} + \sqrt{5}}\)

Step 1: Isolate terms with variable

First, let's separate terms involving the variable \(b\) from constant terms. Move the \(4\) from the right-hand side of the equation to the left-hand side: \[7 - b\sqrt{8} - 4 = b\sqrt{5}\] Simplify the left-hand side: \[3 - b\sqrt{8} = b\sqrt{5}\].

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

\(b = \frac{3}{\sqrt{8} + \sqrt{5}}\)

Key Concepts

Variable Isolation

To solve algebraic equations like \(7-b \sqrt{8}=b \sqrt{5}+4\), the first major step is to isolate variables. In this case, we need to separate all terms related to the variable \(b\) from the constants. This is commonly referred to as 'Variable Isolation'. It helps to clearly see which parts of the equation are affected by the variable. Here's how it works:

• Look at the equation and identify terms with the variable \(b\). In this case, \(-b \sqrt{8}\) and \(b \sqrt{5}\) contain \(b\).
• Our objective is to move constants (like the \(4\) on the right) to one side and variable terms to the other. This makes it easier to focus on what \(b\) can be.
• Shift the constant term to the opposite side by adding or subtracting it from both sides of the equation. By moving \(4\) from the right to the left, we focus on isolating \(b\).

After moving \(4\) from the right side to the left, we get:\[7 - b\sqrt{8} - 4 = b\sqrt{5}\]Which simplifies to:\[3 - b\sqrt{8} = b\sqrt{5}\] Now, \(b\) is neatly separated, leading us to the next step.

Simplifying Expressions

In solving equations, simplifying expressions is crucial. Once we've isolated variable-containing terms from constants, our next objective is to simplify these expressions. This involves performing arithmetic operations like addition, subtraction, and even rationalizing if necessary.With our current equation, \(3 - b\sqrt{8} = b\sqrt{5}\), let's take a closer look:

• Check if the constant terms can be combined or simplified further. Here, no combination is needed as we have already simplified the constant to \(3\).
• Look at \(-b\sqrt{8}\) and \(b\sqrt{5}\). They can be made simpler if there's a common factor, but in this scenario, further simplification involves moving terms rather than direct arithmetic reduction.
• Consider potential manipulation of the equation to simplify sub-expressions. For example, we can add \(b\sqrt{8}\) to both sides if useful in the next steps, but here, we'll focus on the core variable isolation and simplification process.

Since the equation has been carefully organized, we can move on to solving the equation directly. Simplified expressions help in justification and clarity as we solve the equation.

Equation Solving

The final stage in approaching an algebraic equation is solving it. With simplified and isolated terms, we can find the specific values for the variable.In our example, after achieving the equation \(3 - b\sqrt{8} = b\sqrt{5}\), we aim to solve for \(b\). Here's a structured approach to solve it:

• Combine all terms involving the variable \(b\) on one side if not done yet. Here, we already have the left-hand side: \(b\sqrt{5} + b\sqrt{8} = 3\).
• Factor out the variable, if possible. \(b(\sqrt{8} + \sqrt{5}) = 3\).
• Solve for \(b\). To extract \(b\), divide both sides by \(\sqrt{8} + \sqrt{5}\).

Thus,\[b = \frac{3}{\sqrt{5} + \sqrt{8}}\]This requires no additional simplification beyond ensuring that all expressions are accurate. Double-checking your calculation will help verify the correctness. Each step along the way ensures that you accurately solve for the variable, allowing insight into how variables respond under specific conditions. This blueprint will help tackle similar algebraic problems efficiently.