Which of the following is an example of an equation with no solution?

Question

A. 3x + 4 = 3x + 4
B. 3x + 4 = 4x + 3
C. 3x + 4 = 3x + 3
D. 4x + 3 = 3x + 3

Answer 1

An equation with no solution occurs when the variable can be eliminated and results in an impossible statement, such as \(0 = 1\).

Let's analyze each option:

A. \(3x + 4 = 3x + 4\)
This simplifies to \(0 = 0\), which is true for all \(x\). So it has infinitely many solutions.

B. \(3x + 4 = 4x + 3\)
Simplifying this gives:
\[ 3x + 4 - 4x = 3 \]
\[-x + 4 = 3\]
\[-x = -1\]
\[x = 1\]
This has a single solution: \(x = 1\).

C. \(3x + 4 = 3x + 3\)
Simplifying this gives:
\[ 3x + 4 - 3x = 3 \]
\[ 4 = 3 \]
This is an impossible statement (no solution).

D. \(4x + 3 = 3x + 3\)
Simplifying this gives:
\[ 4x - 3x + 3 = 3 \]
\[ x + 3 = 3 \]
\[ x = 0 \]
This has a single solution: \(x = 0\).

Thus, the correct answer is: C. \(3x + 4 = 3x + 3\) (which has no solution).

Answer 2

are you sure this is correct

Answer 3

Yes, I confirm that the analysis is correct. Let's break it down once again to ensure clarity:

1. Option A: \(3x + 4 = 3x + 4\)

   • This simplifies to \(0 = 0\), which is a true statement. This means there are infinitely many solutions (all values of \(x\) are valid).

2. Option B: \(3x + 4 = 4x + 3\)

   • Rearranging gives: \[ 3x + 4 - 4x = 3 \implies -x + 4 = 3 \implies -x = -1 \implies x = 1 \]
   • This equation has a single solution: \(x = 1\).

3. Option C: \(3x + 4 = 3x + 3\)

   • Rearranging gives: \[ 3x + 4 - 3x = 3 \implies 4 = 3 \]
   • This is clearly a false statement (impossible), meaning there are no solutions.

4. Option D: \(4x + 3 = 3x + 3\)

   • Rearranging gives: \[ 4x + 3 - 3x = 3 \implies x + 3 = 3 \implies x = 0 \]
   • This equation has a single solution: \(x = 0\).

So, Option C (\(3x + 4 = 3x + 3\)) is indeed the equation with no solution. Thank you for allowing me to clarify!