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. Start with the given equation: \(\frac{2x}{3} = 6 - \frac{x}{4}\).
2. To eliminate the fractions, find the least common denominator (LCD) of 3 and 4, which is 12. Multiply every term on both sides of the equation by 12 to clear the denominators.
3. After multiplying, simplify each term: \$12 \times \frac{2x}{3} = 12 \times 6 - 12 \times \frac{x}{4}\( becomes \)8x = 72 - 3x$.
4. Next, collect all the variable terms on one side by adding \$3x\( to both sides: \)8x + 3x = 72\( which simplifies to \)11x = 72$.
5. Finally, solve for \(x\) by dividing both sides by 11: \(x = \frac{72}{11}\). After finding \(x\), check the solution by substituting it back into 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 combining like terms, clearing fractions by multiplying both sides by a common denominator, and performing inverse operations such as addition, subtraction, multiplication, or division.

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. Identifying the type depends on the solution set after solving the equation.

Working with Fractions in Equations

Equations with fractions require careful manipulation, often by multiplying both sides by the least common denominator to eliminate fractions. This simplifies the equation and makes it easier to solve for the variable.