Determine whether the equation 3t+4=12+3t has one solution, no solution, or infinitely many solutions.
The Correct Answer and Explanation is :
The equation ( 3t + 4 = 12 + 3t ) has infinitely many solutions. Here’s the step-by-step explanation:
Step 1: Simplify the equation
We begin by simplifying the equation:
[
3t + 4 = 12 + 3t
]
The goal is to isolate ( t ). To do this, let’s start by eliminating the ( 3t ) terms from both sides of the equation.
Step 2: Subtract ( 3t ) from both sides
Subtract ( 3t ) from both sides:
[
3t + 4 – 3t = 12 + 3t – 3t
]
This simplifies to:
[
4 = 12
]
Now, we have a statement that is clearly false (4 does not equal 12).
Step 3: Analyze the result
At this point, we see that after simplifying the equation, we end up with a contradiction: ( 4 = 12 ). This suggests that there is an issue with the equation itself, specifically that the terms involving ( t ) have canceled out and left us with a false statement.
Step 4: Conclusion
The fact that we reached a false statement like ( 4 = 12 ) after simplifying shows that the equation does not hold for any value of ( t ).
This means the equation has no solution because there’s no possible value of ( t ) that will satisfy the equation.
In summary:
- One solution would occur if we could isolate ( t ) and solve for it.
- No solution occurs when simplifying the equation leads to a contradiction, as we see with ( 4 = 12 ).
- Infinitely many solutions would occur if the equation simplified to something true for all values of ( t ), such as ( 0 = 0 ).
In this case, the correct answer is no solution.