Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{3}{2 x-4}-\frac{5}{3 x-6}=\frac{3}{5}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{31}{18} \).

Step 1: Identify a common denominator

The expression on the left consists of two fractions: \( \frac{3}{2x-4} \) and \( \frac{5}{3x-6} \). We need a common denominator to combine these fractions. Notice that \( 2x-4 \) can be factored as \( 2(x-2) \) and \( 3x-6 \) can be factored as \( 3(x-2) \). The common denominator for these fractions is \( 6(x-2) \).

Step 2: Rewrite each fraction with the common denominator

Re-write each fraction so that they have the common denominator \( 6(x-2) \). For \( \frac{3}{2x-4} \), multiply the numerator and denominator by 3 to get \( \frac{9}{6(x-2)} \). For \( \frac{5}{3x-6} \), multiply the numerator and denominator by 2 to get \( \frac{10}{6(x-2)} \).

Step 3: Simplify the left-hand side

Now that both fractions have the same denominator, combine them: \( \frac{9}{6(x-2)} - \frac{10}{6(x-2)} = \frac{9 - 10}{6(x-2)} = \frac{-1}{6(x-2)} \).

Step 4: Solve the equation

Rewrite the equation using the simplified left-hand side: \( \frac{-1}{6(x-2)} = \frac{3}{5} \). Cross-multiply to solve for \( x \): \(-1 \cdot 5 = 3 \cdot 6(x-2)\). This simplifies to \(-5 = 18x - 36\).

Step 5: Isolate x

Add 36 to both sides of the equation to get \( 31 = 18x \). Finally, divide both sides by 18 to solve for \( x \): \( x = \frac{31}{18} \).

Key Concepts

Fractions

Fractions are a way to express numbers that are not whole or natural. They consist of two parts: the numerator at the top and the denominator at the bottom. The numerator tells you how many parts you have, whereas the denominator tells you into how many parts the whole is divided. For instance, in the fraction \( \frac{3}{4} \), there are 3 parts out of a total of 4.When working with fractions in equations, it's crucial to understand how they interact with each other, especially when you need to add, subtract, multiply, or divide them. Fractions can also be daunting because they involve two numbers instead of just one. However, they follow a set of simple rules and properties that, once understood, make manipulating them much easier for solving problems.When performing operations with fractions, especially in equation solving, finding a common denominator or using cross multiplication can simplify the process considerably.

Common Denominator

A common denominator is a shared value between two or more fractions' denominators. Finding a common denominator is essential for adding or subtracting fractions. In the given exercise, to subtract the fractions \( \frac{3}{2x-4} \) and \( \frac{5}{3x-6} \), a common denominator is needed. Here's a simple way to find it:

• Factor each denominator separately. For example, \(2x-4\) can be written as \(2(x-2)\) and \(3x-6\) as \(3(x-2)\).
• Identify common factors in the factored forms. Here, both expressions share \((x-2)\).
• Multiply the unique parts and the common part to get a common denominator. In this case, it becomes \(6(x-2)\) after multiplying \(2 \times 3 \times (x-2)\).

With a common denominator, each fraction can be rewritten with the same base, which simplifies combining or subtracting them, as seen in transforming the fractions in this exercise.

Cross Multiplication

Cross multiplication is a technique used to solve equations where two fractions are set equal to each other. This strategy involves multiplying diagonally across the equal sign to eliminate the fractions, simplifying the problem to a basic algebraic equation.In the given equation\[\frac{-1}{6(x-2)} = \frac{3}{5},\]cross multiplying gives:

• Multiply the numerator of one fraction by the denominator of the other: \(-1 \cdot 5\).
• Multiply the other numerator by the opposite denominator: \(3 \cdot 6(x-2)\).

This gives the equation -5 = 18(x-2), turning an equation with fractions into a straightforward linear equation. Through cross multiplication, you sidestep complex fraction manipulation, allowing easier isolation and calculation of the variable \(x\).

Factoring Expressions

Factoring is the process of breaking down expressions into simpler parts or products. It's essential in simplifying equations, especially when dealing with expressions like polynomials. By breaking down into simple "factors," you can more easily identify common denominators, solutions, or simplify more complex expressions.In this exercise, to find a common denominator, the expressions \(2x-4\) and \(3x-6\) were factored. This conversion revealed the common factor \((x-2)\), simplifying the process of combining the fractions. Factoring can be applied through several methods:

• Factor by grouping, for polynomials that can be split into two pairs.
• Use of distributive law in reverse to factor common terms.
• Recognizing special products like difference of squares.

By focusing on factoring, you not only simplify the equation-solving process but also reveal underlying structures within expressions that might not be apparent at first glance.