Question

NESSENY

Math 8B Unit 4 Study Guide
14. Solve for y to determine if the equation has one, none, or infinite solutions.
by + 24 = 3(2y + 8)
A. This equation has infinite solutions.
B. This equation has no solutions.
C. This equation has one solution: y = 1
D. This equation has one solution: y = 0

Answer 1

To solve the equation \( by + 24 = 3(2y + 8) \), we first expand the right side:

\[
by + 24 = 3 \cdot 2y + 3 \cdot 8
\]
\[
by + 24 = 6y + 24
\]

Next, we can simplify the equation by subtracting \( 24 \) from both sides:

\[
by = 6y
\]

Now, we rearrange to isolate \( y \):

\[
by - 6y = 0
\]
\[
y(b - 6) = 0
\]

From this equation, we can determine solutions by setting \( y = 0 \) or \( b - 6 = 0 \).

1. If \( y = 0 \), that is one potential solution.
2. If \( b - 6 = 0 \), that means \( b = 6 \). In this case, the equation becomes \( 6y = 6y \), which is true for all values of \( y \). Therefore, there are infinite solutions.

If \( b \neq 6 \), then the only solution is \( y = 0 \).

To summarize:
- If \( b = 6 \), there are infinite solutions.
- If \( b \neq 6 \), the only solution is \( y = 0 \).

Without a specific value for \( b \), we cannot definitively choose one of the answer options without knowing if \( b \) is equal to 6 or not. Assuming \( b \neq 6 \) leads us to the solution \( y = 0\).

Thus, among the provided choices, **D. This equation has one solution: y = 0** is correct if we assume \( b \neq 6 \).