To determine how many solutions exist for the given equation:
[tex]\[
6x + 15 = 6(x - 3)
\][/tex]
we will go through a step-by-step process to simplify and analyze the equation.
1. Distribute the 6 on the right side of the equation:
[tex]\[
6x + 15 = 6x - 18
\][/tex]
2. Subtract [tex]\(6x\)[/tex] from both sides to begin isolating the constants:
[tex]\[
6x + 15 - 6x = 6x - 18 - 6x
\][/tex]
which simplifies further to:
[tex]\[
15 = -18
\][/tex]
Now, we encounter a crucial step in our analysis. The equation has reduced to:
[tex]\[
15 = -18
\][/tex]
This is clearly a false statement. Since the simplified form of the equation results in a contradiction (15 does not equal -18), it indicates that there is no value of [tex]\(x\)[/tex] that will satisfy the original equation.
3. Conclusion:
Since the equation leads to a contradiction, it implies that there are no solutions to the equation.
Thus, the correct answer is:
B. 0
