In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4

Verified step by step guidance

1. Multiply every term by the least common denominator (LCD) of the fractions, which is 12, to eliminate the fractions.
2. Distribute the 12 across each term: 12 * ((3x+1)/3) - 12 * (13/2) = 12 * ((1-x)/4).
3. Simplify each term: 4(3x+1) - 6(13) = 3(1-x).
4. Distribute the numbers: 12x + 4 - 78 = 3 - 3x.
5. Combine like terms and solve for x: 12x + 3x = 3 + 78 - 4.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This process typically includes isolating the variable on one side of the equation through operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate both sides of the equation is crucial for arriving at the correct solution.

Types of Equations

Equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity holds true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. Recognizing the type of equation is essential for understanding the implications of the solution found.

Fraction Manipulation

Manipulating fractions is a key skill in algebra, especially when solving equations that involve them. This includes finding a common denominator, simplifying fractions, and performing operations such as addition, subtraction, multiplication, and division. Mastery of these techniques is necessary to accurately solve equations that contain fractional terms.