To determine which equation has no solution, we can solve each equation step-by-step to see if we arrive at a contradiction.
Let's evaluate each equation:
Option A:
\[ 15z + 12 = 5x + 4 \]
Rearranging, we get:
\[ 15z - 5x = -8 \]
This can potentially have many solutions depending on the values of \(z\) and \(x\).
Option B:
\[ 8 - 4z = 10 - 4 + 2x \]
This simplifies to:
\[ 8 - 4z = 6 + 2x \]
Rearranging gives:
\[ -4z - 2x = -2 \]
Which can again have solutions depending on \(z\) and \(x\).
Option C:
\[ 43 - 20 = 2x - 10 \]
This simplifies to:
\[ 23 = 2x - 10 \]
Adding 10 to both sides:
\[ 33 = 2x \]
Dividing by 2, we get:
\[ x = 16.5 \]
This has one solution.
Option D:
\[ 15 + 5 = 5x + 6 \]
This simplifies to:
\[ 20 = 5x + 6 \]
Rearranging gives:
\[ 14 = 5x \]
Dividing by 5, we get:
\[ x = \frac{14}{5} \]
This also has a solution.
Next, let's look for contradictions or cases where the equations lead to impossible statements.
In options A, B, C, and D, we do not find any of them leading directly to a contradiction but we still need to make sure about each:
- Option A and Option B: Both have the form where they relate variables and could have solutions.
- Option C gives a specific solution.
- Option D also leads to a specific solution.
Upon re-evaluation, it seems all provided equations could have solutions. However, if any of these could be contrived to show no solutions through contradictory values, then there may be an oversight.
Conclusion:
If looking specifically for contradictions in terms of values such as equal values leading to false statements such as \(z = z + c\) where \(c \neq 0\), none are immediately visible.
Could you re-check if there may be more specific errors in how the original equations were constructed or their forms? Currently from the analysis, none provide a definitive fail case without further context on variable constraints.