Do -5x+12=-5x-12 have exactly one solutions
The Correct Answer and explanation is :
The correct answer is: No, the equation does not have exactly one solution; it has no solutions.
To determine whether the equation (-5x + 12 = -5x – 12) has exactly one solution, we can start by simplifying and analyzing the equation.
Step 1: Rearranging the Equation
The equation can be rewritten as:
[
-5x + 12 + 5x = -5x – 12 + 5x
]
This simplifies to:
[
12 = -12
]
Step 2: Analyzing the Result
The equation (12 = -12) is a contradiction because (12) is not equal to (-12). This means that the original equation has no solutions at all.
Step 3: General Consideration of Linear Equations
In general, a linear equation of the form (Ax + B = Cx + D) can have:
- Exactly one solution: This occurs when (A) and (C) are different, leading to a unique value of (x) that satisfies the equation.
- No solution: This happens when you arrive at a contradiction, such as (k = m) where (k \neq m), as we found here.
- Infinitely many solutions: This is the case when the equation simplifies to a tautology, such as (k = k).
Conclusion
In this case, since we simplified the equation and found that it results in a false statement, we conclude that the equation (-5x + 12 = -5x – 12) does not have exactly one solution; rather, it has no solutions at all. This illustrates an important concept in algebra: contradictions arise when attempting to equate two expressions that are inherently unequal. Understanding these principles helps in solving equations and identifying their characteristics effectively.
Thus, the answer is: No, the equation does not have exactly one solution; it has no solutions.