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. Start by writing down the given equation: \(\frac{3x+1}{3} - \frac{13}{2} = \frac{1 - x}{4}\).
2. To eliminate the fractions, find the least common denominator (LCD) of 3, 2, and 4, which is 12. Multiply every term on both sides of the equation by 12 to clear the denominators.
3. Distribute 12 to each term: \(12 \times \frac{3x+1}{3} - 12 \times \frac{13}{2} = 12 \times \frac{1 - x}{4}\).
4. Simplify each term after multiplication: \(4(3x+1) - 6(13) = 3(1 - x)\).
5. Now, expand the parentheses and solve the resulting linear equation for \(x\). After finding \(x\), check if the solution satisfies the original equation to determine if it is an identity, conditional, or inconsistent equation.

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. This often requires clearing fractions by finding a common denominator or multiplying both sides by the least common multiple. Understanding how to manipulate equations step-by-step is essential to find the solution accurately.

Types of Equations: Identity, Conditional, and Inconsistent

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. Recognizing these types helps interpret the solution set after solving the equation, indicating whether the equation holds universally, sometimes, or never.

Working with Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. Solving equations involving them requires careful handling of denominators to avoid division by zero and to simplify expressions correctly. Clearing denominators by multiplying through is a common strategy to simplify the equation.