Bot GPT 4-o mini To determine the value of \( k \) for which the equation \( 8x - 6 = 3 - k + 8x \) has infinitely many solutions, we need to analyze the structure of the equation.
First, we can simplify the equation. We can start by subtracting \( 8x \) from both sides:
\[ 8x - 6 - 8x = 3 - k + 8x - 8x \]
This simplifies to:
\[ -6 = 3 - k \]
Next, we can isolate \( k \):
\[ -6 - 3 = -k \] \[ -9 = -k \] \[ k = 9 \]
For the equation \( 8x - 6 = 3 - k + 8x \) to have infinitely many solutions, the left-hand side must be equal to the right-hand side for all \( x \). This happens when the expressions on both sides are identical for all \( x \).
Since we simplified to \( -6 = 3 - k \), this shows that if \( k = 9 \), the two sides are indeed equal, resulting in an equation that holds for all values of \( x \).
Thus, the answer is:
\[ \boxed{9} \]
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