Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

There is no solution because the equation simplifies to a false statement.

Step 1: Simplify Fractions

Let's start by simplifying the fractions in the given equation. Notice that the denominator of the second fraction can be factored: \[\frac{6}{4x + 6} = \frac{6}{2(2x + 3)} = \frac{3}{2x + 3}\]Now, substitute this back into the equation: \[\frac{2x}{2x + 3} + \frac{3}{2x + 3} = 5\]

Step 2: Combine Like Terms

Since both fractions have the same denominator, we can combine the numerators:\[\frac{2x + 3}{2x + 3} = 5\]

Step 3: Simplify the Equation

The fraction can be simplified because the numerator and the denominator are the same:\[1 = 5\]

Step 4: Analyze the Result

The equation simplifies to \(1 = 5\), which is a contradiction. This implies that there is no value of \(x\) that can satisfy the original equation.

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. In the given exercise, the fraction \( \frac{6}{4x + 6} \) is simplified. By factoring the denominator, we find that \( 4x + 6 = 2(2x + 3) \). Hence, the fraction becomes \( \frac{6}{2(2x + 3)} \) and simplifies to \( \frac{3}{2x + 3} \).

This simplification makes it easier to work with the equation, as it prepares the fractions for operations like addition or subtraction. Reducing fractions also allows for clearer observation of relationships between numerators and denominators, facilitating the solving process.

Like Terms

Like terms are terms that have the same variable raised to the same power. In algebra, combining like terms is crucial for simplifying expressions and solving equations.

In our exercise, both fractions, \( \frac{2x}{2x + 3} \) and \( \frac{3}{2x + 3} \), have the same denominator \( 2x + 3 \). This allows us to combine them easily by adding their numerators: \( 2x + 3 \).

The sum, \( \frac{2x + 3}{2x + 3} \), conveniently reduces the expression since the denominators are identical, leading to a simplified form which directly impacts the equation's next steps.

Contradiction in Equations

A contradiction in an equation occurs when the solution obtained is logically impossible. Once we simplified the fractions and combined like terms, the expression results in \( \frac{2x + 3}{2x + 3} = 5 \). This simplifies further to \( 1 = 5 \), a clear contradiction.

Such a contradiction means there is no possible value of \( x \) that can satisfy the equation. In algebra, recognizing contradictions is crucial as it signifies that the initial equation does not have a valid solution within the realm of real numbers.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. The process often includes simplifying expressions, combining like terms, and isolating the variable.

In this exercise, we followed a path of simplification and looked for a solution, but encountered a contradiction instead. Understanding each step—from simplification to recognition of contradictions—helps in addressing more complex scenarios and solidifying algebraic problem-solving skills.

• Recognize similar terms and simplify fractions first.
• Combine like terms to simplify the overall structure of the equation.
• Look out for contradictions which reveal no solution exists.

Mastering these steps is essential for effectively tackling linear equations in algebra.