Identify the equation as a conditional equation, contradiction, or identity.
[tex]3 p+4=3 p[/tex]
A. The equation is a conditional equation.
B. The equation is a contradiction.
C. The equation is an identity.
Answer (2)
The equation 3 p + 4 = 3 p is identified as a contradiction because it simplifies to the false statement 4 = 0 , which holds true for no value of p .
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Subtract 3 p from both sides of the equation: 3 p + 4 − 3 p = 3 p − 3 p .
Simplify the equation: 4 = 0 .
Since 4 = 0 is a false statement, the equation is a contradiction.
The equation is a contradiction .
Explanation
Understanding the Problem We are given the equation 3 p + 4 = 3 p . Our goal is to determine whether this equation is a conditional equation, a contradiction, or an identity. A conditional equation is true for only some values of the variable. A contradiction is never true, no matter what value is substituted for the variable. An identity is always true, no matter what value is substituted for the variable.
Subtracting 3p from Both Sides To determine the nature of the equation, we will try to simplify it by performing algebraic operations on both sides. We start by subtracting 3 p from both sides of the equation: 3 p + 4 − 3 p = 3 p − 3 p
Simplifying the Equation This simplifies to: 4 = 0
Identifying the Equation Type The equation 4 = 0 is a statement that is always false, regardless of the value of p . Therefore, the given equation is a contradiction.
Final Answer The equation 3 p + 4 = 3 p is a contradiction because it simplifies to 4 = 0 , which is never true. Therefore, the correct answer is: The equation is a contradiction.
Examples
Consider a scenario where you're trying to balance a budget. If you have an equation that represents your income and expenses, and that equation turns out to be a contradiction (like 4 = 0 ), it means there's no possible way to balance your budget with the current income and expenses. This understanding helps you realize the need to adjust either your income or your expenses to achieve a balanced budget. Recognizing contradictions in mathematical models can highlight impossible scenarios in real-world situations, prompting necessary adjustments or re-evaluations.
