In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2x/3 = 6 - x/4

Verified step by step guidance

1. Multiply every term by the least common denominator (LCD) to eliminate the fractions. The LCD of 3 and 4 is 12.
2. Multiply each term by 12: \(12 \cdot \frac{2x}{3} = 12 \cdot 6 - 12 \cdot \frac{x}{4}\).
3. Simplify each term: \(4 \cdot 2x = 72 - 3x\).
4. Combine like terms by adding \(3x\) to both sides: \(4 \cdot 2x + 3x = 72\).
5. Simplify the equation to solve for \(x\).

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This typically requires isolating the variable on one side of the equation through various algebraic operations, such as addition, subtraction, multiplication, and division. In the given equation, the goal is to manipulate the terms to solve for 'x'.

Types of Equations

Equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. Understanding these classifications helps in determining the nature of the solution after solving the equation.

Fraction Manipulation

Fraction manipulation is essential in solving equations that involve fractions, as seen in the given problem. This includes finding a common denominator, simplifying fractions, and performing operations on both sides of the equation to eliminate fractions. Mastery of these skills is crucial for accurately solving equations and avoiding errors.