$3p + 4 = 3p$

Part 1 of 2

Is the equation a conditional equation, a contradiction, or an identity?

Part 2 of 2

What is the solution set?

The equation 3 p + 4 = 3 p is a contradiction, as simplifying it leads to the false statement 4 = 0 . Therefore, the solution set is empty, represented as { } .
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Answered by Anonymous | 2025-07-04

Subtract 3 p from both sides of the equation: 3 p + 4 − 3 p = 3 p − 3 p .
Simplify the equation to 4 = 0 .
Since 4 = 0 is a false statement, the equation is a contradiction.
The solution set is the empty set: { } ​ .

Explanation

Understanding the Problem We are given the equation 3 p + 4 = 3 p . Our goal is to classify this equation as either a conditional equation, a contradiction, or an identity, and then to determine its solution set.

Simplifying the Equation To classify the equation, we will simplify it by subtracting 3 p from both sides:


3 p + 4 − 3 p = 3 p − 3 p
This simplifies to:
4 = 0

Classifying the Equation The resulting equation, 4 = 0 , is a false statement. This means that there is no value of p that will make the original equation true. Therefore, the equation is a contradiction.

Determining the Solution Set Since the equation is a contradiction, there are no solutions. The solution set is the empty set, which can be represented as { } .

Final Answer Therefore, the equation 3 p + 4 = 3 p is a contradiction, and its solution set is the empty set.


Examples
Consider a scenario where you are trying to balance a budget. If you have an equation that represents your income and expenses, and simplifying the equation leads to a contradiction (like 4 = 0 ), it means that your budget is fundamentally unbalanced, and there is no amount of adjustment that can make your income equal to your expenses. This understanding helps you recognize that you need to either increase your income or decrease your expenses to achieve a balanced budget. This type of problem is a linear equation, and we solve it using algebraic manipulation.

Answered by GinnyAnswer | 2025-07-04