Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{x}{0.6321}+\frac{x}{0.0692}=1000\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{0.6321}+\frac{x}{0.0692}=1000\) is \(x = 62.379\).

Step 1: Combine Like Terms

The equation can be rewritten as \(x(\frac{1}{0.6321}+\frac{1}{0.0692}) = 1000\). This combines the like terms, which in this case, are the terms involving \(x\). Divide \(1\) by \(0.6321\) and \(0.0692\) respectively to simplify.

Step 2: Calculate Coefficients

Now, perform the division \(\frac{1}{0.6321}\) and \(\frac{1}{0.0692}\) respectively. Add these two values together to get \(1.5813609467455622 + 14.44979919678715\).

Step 3: Simplify the Equation

Addition of the above two calculated values gives \(16.031160143532716\). So now, our equation will look like: \(16.031160143532716 \times x = 1000\).

Step 4: Solve for \(x\)

Rearrange the equation to solve for \(x\): \(x = \frac{1000}{16.031160143532716}\).

Step 5: Final Simplification

Perform the division operation to get value of \(x\) rounded to three decimal places: \(x = 62.379\)

Key Concepts

Rational Equations

Rational equations are equations that involve fractions of polynomial expressions. In the exercise, the equation \( \frac{x}{0.6321} + \frac{x}{0.0692} = 1000 \) is a rational equation because \( x \) appears in the numerator of fractions. Such equations require careful manipulation of the fractions involved. To solve these equations, one common method is to combine like terms, bringing similar terms together. This simplifies the equation and makes it easier to handle with a calculator. For instance, factor out the common variable \( x \) from the fractions as shown in the solution steps. Working with rational equations often involves:- Combining terms with shared variables- Finding a common denominator - Simplifying complex fractions into simpler expressionsThis manipulative process is key in managing rational equations effectively.

Use of Calculators

Using a calculator is sometimes essential when solving equations, especially those involving precise decimal numbers. Calculators help to perform repetitive, cumbersome calculations efficiently.For the given exercise, calculators especially come in handy for:

• Dividing precise decimal numbers like \( \frac{1}{0.6321} \) and \( \frac{1}{0.0692} \), saving time and ensuring accuracy.
• Adding calculations together accurately. This example involves adding values such as \( 1.58136 \) and \( 14.4498 \), which might get tricky manually.
• Rounding numbers accurately. Calculators can round off decimals to a desirable precision, such as three decimal places for the final solution.

Using a calculator's precision ensures that solutions are accurate and reduces human error in lengthy algebraic operations.

Numerical Approximation

Numerical approximation entails finding a value close to the actual solution but not necessarily exact. In the context of solving algebraic equations, rounding numbers is often necessary.The original exercise instructed rounding the solution to three decimal places. This practice, though it introduces some level of approximation, helps in expressing belief in the solution's preciseness without overcomplicating the representation.Here's how numerical approximation fits in:

• After obtaining the complex number \( x = \frac{1000}{16.03116014} \), it's more practical to express \( x \) in a rounded form.
• Approximating to \( x = 62.379 \) allows for an understandable and manageable result without losing essential accuracy.

Numerical approximation is a powerful tool in various mathematical solutions, ensuring solutions remain sensible while maintaining their integrity.

Problem-Solving

In algebra, problem-solving involves a series of logical steps that clarify what the problem is asking and how best to approach the solution. The goal is to arrive at a solution that answers the question posed by the equation.To effectively solve a rational equation like the one in the exercise, you need to:

• Understand what the question requires—here, finding the value of \( x \) that satisfies the equation.
• Break down complex steps into manageable parts. Start by rewriting the equation to combined terms, simplifying fractions, calculating coefficients, and finally solving for \( x \).
• Verify the solution by ensuring it fits the original equation context.

Each step in problem-solving builds on the previous, and through a combination of logical reasoning and mathematical manipulation, the desired result is achieved. Always ensure to check your solutions to validate their correctness and applicability to the problem context.