Problem 90 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(\frac{a+1}{b}=\frac{a-1}{b}+\frac{b+1}{a} ; \quad\) for \(a\)

Step-by-Step Solution

Answer

The solution for \(a\) is \(a = \frac{b^2 + b}{2}\).

Step 1: Simplify Both Sides

First, we notice both sides of the equation have fractions. Let's simplify them to start with a common denominator. The equation is \( \frac{a+1}{b} = \frac{a-1}{b} + \frac{b+1}{a} \). Multiply all terms by \( ab \) to eliminate the denominators, yielding:\[ ab \cdot \frac{a+1}{b} = ab \cdot \frac{a-1}{b} + ab \cdot \frac{b+1}{a} \]This simplifies to:\[ a(a+1) = a(a-1) + b(b+1) \]

Step 2: Expand Terms

Expand both sides of the equation:On the left side: \[ a(a+1) = a^2 + a \]On the right side: \[ a(a-1) + b(b+1) = a^2 - a + b^2 + b \]So, the equation becomes:\[ a^2 + a = a^2 - a + b^2 + b \]

Step 3: Combine Like Terms

Now, let's combine like terms.Subtract \(a^2\) from both sides:\[ a = -a + b^2 + b \]Add \(a\) to both sides:\[ 2a = b^2 + b \]

Step 4: Solve for 'a'

To solve for \(a\), divide both sides by 2:\[ a = \frac{b^2 + b}{2} \]

Step 5: Verify the Solution

It's always a good practice to verify your solution. Substitute \( a = \frac{b^2 + b}{2} \) back into the original equation and check if both sides equal.Replace \(a\) in \( \frac{a+1}{b} = \frac{a-1}{b} + \frac{b+1}{a} \), and algebraically verify that the equality holds. This confirms the correctness of the solution.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a process used to rearrange and simplify equations, making them easier to solve. The main goal is to perform operations that help isolate the variable of interest or simplify complex expressions. In the given exercise, algebraic manipulation helps transform the fractions in the equation into a simpler form. By multiplying each term by the common denominator (in this case, \(ab\)), we eliminate the fractions altogether.

• This makes the equation less complex and more straightforward.
• We then expand and simplify these newly formed terms without fractions.

Understanding how to manipulate equations is crucial because it forms the backbone of solving algebraic problems. It involves skills like recognizing common patterns, knowing which operations to perform, and deciding when to apply certain techniques.

Fractions

Fractions can often make an equation look more complicated because they involve division. In our equation, we have different fractions with variables as numerators and denominators. They represent parts of a whole or division of quantities. Understanding how to work with fractions is essential in algebra because:

• They allow us to work with numbers that aren't whole, offering more precise solutions.
• Fractions require careful mathematical operations like addition, subtraction, multiplication, and division, with specific rules.

When solving algebraic equations with fractions, it's important to either find a common denominator or eliminate these fractions by multiplying each term by the least common multiple of the denominators. This simplifies the equation's complexity, making it easier to proceed to subsequent steps.

Common Denominator

A common denominator is a shared multiple of the denominators in an equation involving fractions. Identifying a common denominator is a crucial step because:

• It allows us to add or subtract fractions directly without altering their values.
• By converting fractions so they have the same denominator, we can effectively combine them as needed.

In our exercise, we observed fractions with different denominators. To simplify:- We multiplied the entire equation by \(ab\), the product of the individual denominators.- This step was fundamental in eliminating the fractional parts and allowed us to combine them into polynomial expressions that were easier to manipulate.Thus, finding a common denominator or removing it altogether simplifies fraction-heavy equations immensely.

Variable Isolation

Variable isolation refers to the process of manipulating an equation so that a specific variable stands alone on one side of the equation. This is the main goal in solving equations, as it reveals the value of the variable. In the provided exercise:

• We first made sure to remove fractions by using a common denominator.
• We expanded polynomials and simplified the equation by combining like terms.

Finally, by performing simple algebraic operations, we isolated \(a\) and expressed it in terms of \(b\): \( a = \frac{b^2 + b}{2} \). Achieving this:- Ensures the problem's requirements are met as set by the problem statement.- Provides a clear, understandable solution that is easy to check and verify.Variable isolation is significant in algebra because it directly leads to the solution of equations, allowing us to find exact values or conditions for which an equation holds true.