Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 2(x - 8) = 3x - 16

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Start by expanding the left side of the equation: multiply 2 by each term inside the parentheses to get $2(x - 8) = 2x - 16$.

Rewrite the equation with the expanded left side: $2x - 16 = 3x - 16$.

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$.

Then, isolate $x$ by adding 16 to both sides: $-16 + 16 = x - 16 + 16$, which simplifies to $0 = x$.

Interpret the result: since $x = 0$ is the only solution, the equation is a conditional equation with the solution set $\{0\}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Types of Equations: Identity, Conditional, and Contradiction

An identity is an equation true for all values of the variable, a conditional equation is true only 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.

Checking Solutions and Solution Sets

After solving, substituting the solution back into the original equation verifies its validity. The solution set includes all values that satisfy the equation, which can be a single value, all real numbers, or an empty set.