Problem 32 - Technical Mathematics with Calculus

Question

Literal Fractional Equations. $$\frac{a-b}{b x+c}+\frac{a+b}{a x-c}=0$$

Step-by-Step Solution

Answer

The solution involves finding a common denominator, simplifying the equation, and solving for x.

Step 1: Combine Fractions

Multiply both sides of the equation by the common denominator \(b x+c\)\(a x-c\) to eliminate the fractions.

Step 2: Expand and Simplify

Expand the equation by distributing terms across the brackets and simplify the resulting expressions.

Step 3: Factor and Solve for x

Factor any common terms in the resulting equation. Then, solve for x by isolating it on one side of the equation.

Key Concepts

Combine Fractions

When working with literal fractional equations, combining fractions is often the first step to solving the problem. Imagine having pieces of a pizza divided across different boxes; to know the complete picture, you need to bring all the pieces together. Similarly, with fractions, you need to find a common denominator to 'combine' them into a single expression.

For the equation \(\frac{a-b}{b x+c}+\frac{a+b}{a x-c}=0\), the common denominator is \(b x+c)(a x-c)\). Multiplying both sides of the equation by this common denominator allows us to 'combine' the fractions by getting rid of the fraction form altogether, transforming the problem into a polynomial equation—one step closer to finding the value of x.

Expand and Simplify Expressions

With the fractions combined, the next step is to expand and simplify the expressions. This is akin to unpacking a suitcase to see what's inside; we distribute the terms and then tidy up. Using the distributive property, we expand expressions by multiplying each term in the first bracket by each term in the second bracket.

For example, multiplying \(a-b)\) by \(b x+c)\) and \(a+b)\) by \(a x-c)\) results in a longer expression that may look overwhelming at first sight. However, simplifying the expression by combining like terms—the ones with the same variables and exponents—helps tidy our mathematical 'room'. The equation becomes neater, often revealing a clearer path toward the solution.

Factor and Solve Equations

After expansion and simplification, our next act is to factor the equation. Factoring is like finding the original building blocks of a structure; it's about recognizing patterns and breaking down the complex expression into simpler, multiplicative components.

Factors are expressions that multiply together to give the original expression. Once the equation from the previous steps is boiled down to its factors, solving for x becomes much more straightforward. It's like solving a puzzle—when you know the right pieces and where they fit, the puzzle's picture (in this case, the value of x) becomes clear. By setting each factor equal to zero and solving for x, we find the possible solutions to the initial fractional equation.