Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{-5}{3 x-9}+\frac{4}{x-3}=\frac{5}{6}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{29}{5} \).

Step 1: Identify a Common Denominator

To solve the equation \( \frac{-5}{3x-9} + \frac{4}{x-3} = \frac{5}{6} \), we first need to recognize that the expression \(3x-9\) can be written as \(3(x-3)\). So the equation becomes \( \frac{-5}{3(x-3)} + \frac{4}{x-3} = \frac{5}{6} \). The least common denominator (LCD) here is \(3(x-3)\).

Step 2: Rewrite Fractions with the Common Denominator

We need to express each fraction with the LCD: \( \frac{-5}{3(x-3)} + \frac{12}{3(x-3)} = \frac{5}{6} \). Notice the second fraction was multiplied by 3 in both the numerator and the denominator, transforming \( \frac{4}{x-3} \) into \( \frac{12}{3(x-3)} \).

Step 3: Combine the Fractions

Combine the fractions on the left-hand side of the equation: \( \frac{-5 + 12}{3(x-3)} = \frac{5}{6} \). This simplifies to \( \frac{7}{3(x-3)} = \frac{5}{6} \).

Step 4: Cross-Multiply and Solve for x

Cross-multiply to eliminate the fractions: \( 7 imes 6 = 5 imes 3(x-3) \). This results in the equation \( 42 = 15(x-3) \).

Step 5: Expand and Simplify

Expand and simplify the equation: \( 42 = 15x - 45 \). Add 45 to both sides to get \( 87 = 15x \).

Step 6: Solve for x

Divide both sides by 15 to isolate \( x \): \( x = \frac{87}{15} \). Simplify the fraction to \( x = \frac{29}{5} \).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and rules for manipulating those symbols. It helps us describe relationships using variables and constants. In algebra, we often work with expressions, equations, and understand the relationships between numbers.

An equation is a mathematical statement that asserts the equality of two expressions. Solving an algebraic equation involves finding the values of the variables that satisfy this equality. In our example, the equation is given with fractions, showcasing the need for algebraic manipulation to find the solution.

Understanding algebra is crucial as it lays the groundwork for more advanced topics in mathematics. By learning algebra, you gain skills in logical thinking and problem-solving, which are applicable to everyday life.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they contain algebraic expressions in their numerators or denominators. These expressions include variables that need to be handled with care when solving equations.

• A fraction like \( \frac{4}{x-3} \) is called an algebraic fraction because the denominator contains the variable \( x \).
• To solve equations involving algebraic fractions, it's important to find a common denominator, which allows us to combine and manipulate the fractions more easily.

In our exercise, you are given algebraic fractions on both sides of an equation. This requires you to rewrite them with a common denominator so they can be combined. Understanding how to work with algebraic fractions is a key part of solving rational equations.

Equation Solving

Equation solving involves finding the value of unknown variables that satisfy the given equation. When faced with an equation containing fractions, the process can become more complex, but the principles remain the same:

• Identify all components of the equation, including fractions and other terms.
• Find a common denominator to handle the fractions easily.
• Use algebraic manipulation techniques, such as combining like terms, cross-multiplying, and simplifying expressions.

In our step-by-step solution, after rewriting the fractions with a common denominator, we first simplified the equation, and then solved for the variable \( x \) by isolating it on one side. This linear algebra technique is essential for solving rational equations efficiently.

Least Common Denominator

The least common denominator (LCD) is the smallest multiple that is shared by the denominators of two or more fractions. Finding the LCD is crucial when you need to add, subtract, or compare fractions with different denominators.

• In the equation \( \frac{-5}{3x-9} + \frac{4}{x-3} = \frac{5}{6} \), the expressions \(3x-9\) and \(x-3\) needed a common denominator to be combined.
• Here, recognizing that \(3x-9\) can be rewritten as \(3(x-3)\) helps in identifying the LCD as \(3(x-3)\).

Using the LCD, fractions can be rewritten and combined easily. It simplifies complex equations and is a fundamental skill in working with algebraic fractions. Mastering the concept of LCD enables smoother transition when performing operations on rational expressions.