Problem 75 - Precalculus

Question

Given the equation \(\frac{1}{x}=\frac{1}{a}+\frac{1}{b}\) (a) Solve to show \(x=\frac{a b}{a+b},\) provided \(a+b \neq 0\) (b) Check the solution.

Step-by-Step Solution

Answer

The solution is correct: x = \(\frac{ab}{a+b}\).

Step 1: Write the Equation in a Single Fraction Form

Start with the given equation: \( \frac{1}{x} = \frac{1}{a} + \frac{1}{b} \). The goal is to combine the right-hand side into a single fraction. Using the common denominator \(ab\), we can rewrite the equation as \( \frac{1}{x} = \frac{b}{ab} + \frac{a}{ab} \).

Step 2: Combine the Fractions

Add the fractions on the right-hand side: \( \frac{1}{x} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab} \). Now the equation is \( \frac{1}{x} = \frac{a+b}{ab} \).

Step 3: Solve for x

To solve for \(x\), take the reciprocal of both sides of the equation from Step 2, yielding \( x = \frac{ab}{a+b} \). This assumes \(a + b eq 0\) to avoid division by zero.

Step 4: Check the Solution

Substitute \(x = \frac{ab}{a+b}\) back into the original equation to verify: The left-hand side becomes \(\frac{1}{x} = \frac{a+b}{ab}\). The right-hand side \(\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}\), confirming that both sides are equal, thus verifying our solution.

Key Concepts

Fraction Simplification

Fractions often appear complex, but they can be made much simpler through proper simplification techniques, which is a fundamental concept in algebraic equations. Simplifying a fraction involves rewriting it so that both the numerator and the denominator are as small as possible, without changing the value of the fraction.
\[ \frac{1}{a} + \frac{1}{b} \]
becomes
\[ \frac{b}{ab} + \frac{a}{ab} \ = \frac{a+b}{ab} \]
Here, both fractions have been rewritten with a common denominator \(ab\). This makes adding them much easier because now it's a simple case of adding the numerators, which gives us a single simplified fraction.
Understanding fraction simplification is a powerful tool because it allows us to manipulate complex algebraic expressions into simpler forms. This skill is essential for solving equations as it often reveals hidden relationships and makes computations more straightforward.

Reciprocals

The concept of reciprocals plays a crucial role in solving algebraic equations, especially when dealing with fractions. The reciprocal of a number or a fraction is simply one divided by that number. For example, the reciprocal of \(x\) is \(\frac{1}{x}\), and similarly, the reciprocal of \(\frac{1}{x}\) is \(x\).
In our original equation \(\frac{1}{x} = \frac{a+b}{ab}\), solving for \(x\) involves taking the reciprocal of both sides:
\[ x = \frac{ab}{a+b} \]
This step of flipping the fractions—taking reciprocals—is key to deriving the final answer. This method allows us to effectively solve equations that include fractions, by changing the focus from division to multiplication, which is often easier to handle.
Reciprocals are particularly useful in algebra because they allow us to "move" a variable or term out of a denominator, essentially enabling us to eliminate division from the equation. This makes it easier to visualize and carry out the solution, confirming the simplicity and elegance that reciprocals provide.

Verification of Solutions

Once you've solved an equation, it's important to verify the solution to ensure that the answer is indeed correct. This step involves substituting the solution back into the original equation to check if both sides are equal.
Let's suppose we found \(x = \frac{ab}{a+b}\) as a solution. To verify, substitute this back into the original equation \(\frac{1}{x} = \frac{1}{a} + \frac{1}{b}\).
Here’s what we do:

• On the left side, replace \(x\) with \(\frac{ab}{a+b}\), giving \(\frac{1}{x} = \frac{a+b}{ab}\).
• Calculate the right side, \(\frac{1}{a} + \frac{1}{b}\), which we simplified earlier to \(\frac{a+b}{ab}\).

Both sides match, thus verifying the accuracy of our solution.
Verification is essential to confirm that no errors were made in the solution process and is a step that should always be performed. It's the final check that supports your conclusion, acting as a mathematical guarantee that your problem-solving method was correct.