Problem 86 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ \frac{a+1}{b}=\frac{a-1}{b}+\frac{b+1}{a} ; \text { for } a $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is \( \frac{a+1}{b} = \frac{a-1}{b} + \frac{b+1}{a} \), and we need to solve for the variable \( a \). This means that we need to manipulate the equation to express \( a \) alone on one side.

Step 2: Eliminate Fractions

To eliminate the fractions, multiply through by the common denominator \( ab \). This gives: \[ a(a+1) = a(a-1) + b(b+1) \] Simplifying, you have: \[ a^2 + a = a^2 - a + b^2 + b \]

Step 3: Simplify the Equation

Simplify the equation by subtracting \( a^2 \) from both sides of the equation:\[ a = -a + b^2 + b \]Then, move all terms involving \( a \) to one side:\[ 2a = b^2 + b \]

Step 4: Solve for a

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

Key Concepts

Variables in Fractions

When dealing with equations involving fractions, it's common to find variables either in the numerator, the denominator, or both. In the original exercise, the variable \( a \) is present in two fractions: \( \frac{a+1}{b} \) and \( \frac{b+1}{a} \). Understanding how these variables interact in fractions is crucial.

• Variables in the numerator: They are directly multiplied by the denominator when attempting to eliminate fractions.
• Variables in the denominator: Finding a common denominator helps in simplifying calculations and is essential for eliminating the fractions.

This can make the manipulation of these equations more complex because any change affects how each fraction behaves.

Eliminating Fractions

One effective strategy to simplify equations with fractions is to eliminate the fractions by multiplying both sides by a common denominator. This makes the equation much simpler and easier to manage. In the given problem, the fractions were removed by using the common denominator \( ab \). This step turns the fractional equation into a polynomial equation:

• Multiply every term by the common denominator.
• This results in removing the fractions, simplifying the problem greatly.
• Be mindful to distribute correctly when using the common denominator.

Eliminating fractions makes subsequent algebraic manipulation less daunting and much more straightforward.

Manipulating Equations

Once fractions are eliminated, the next step involves manipulating the equation to isolate the variable of interest. For example, in the original task, simplifying \( a(a+1) = a(a-1) + b(b+1) \) after eliminating fractions gives you the equation \( a^2 + a = a^2 - a + b^2 + b \).

• Cancel out like terms to simplify the equation.
• Move terms involving the variable to one side to start the isolation process.

Proper manipulation means strategically moving terms across the equation, while maintaining equality and simplifying at each step.

Algebraic Simplification

Simplifying algebraic expressions is essential, especially after manipulating the equation to isolate the variable. In the simplified equation \( 2a = b^2 + b \), simplification involves solving for \( a \), which is done by dividing both sides by 2:

• Divide the equation’s terms around the variable by the coefficient of the variable.
• Check each step to ensure simplifications maintain the equality.

By following these steps, you reach the solution \( a = \frac{b^2 + b}{2} \). Understanding algebraic simplification helps avoid common mistakes and allows you to handle complex expressions efficiently.