Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. 2(x-8) = 3x-16

Verified step by step guidance

1. Start by expanding the left side of the equation: \$2(x-8)\( becomes \(2 \cdot x - 2 \cdot 8\), which simplifies to \)2x - 16$.
2. Rewrite the equation with the expanded left side: \$2x - 16 = 3x - 16$.
3. Next, isolate the variable terms on one side by subtracting \$2x\( from both sides: \)2x - 16 - 2x = 3x - 16 - 2x\(, which simplifies to \)-16 = x - 16$.
4. Then, isolate \(x\) by adding 16 to both sides: \(-16 + 16 = x - 16 + 16\), which simplifies to \$0 = x$.
5. Interpret the result: since \(x = 0\) is a specific solution, the equation is a conditional equation with the solution set \(\{0\}\).

Key Concepts

Types of Equations: Identity, Conditional, and Contradiction

An identity is an equation true for all values of the variable, a conditional equation is true for specific values, and a contradiction has no solution. Recognizing these types helps determine the nature of the solution set.

Solving Linear Equations

Solving linear equations involves isolating the variable by applying inverse operations such as addition, subtraction, multiplication, or division. This process helps find the values that satisfy the equation.

Solution Set of an Equation

The solution set is the collection of all values that make the equation true. It can be a single value, all real numbers, or empty, depending on whether the equation is conditional, an identity, or a contradiction.