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The given equation is:

\[
6x + 15 = a(x + b)
\]

To find the value of \( b \) when the equation has infinitely many solutions, the left-hand side (LHS) and right-hand side (RHS) must represent the same line. For this to happen, both the coefficients of \(x\) and the constants must be equal on both sides.

### Step 1: Expand the right-hand side
Expand \(a(x + b)\):

\[
a(x + b) = ax + ab
\]

So, the equation becomes:

\[
6x + 15 = ax + ab
\]

### Step 2: Equate the coefficients of \(x\) and the constants
For the equation to have infinitely many solutions, the coefficients of \(x\) and the constants must be equal on both sides:

1. Coefficient of \(x\): \(6 = a\)
2. Constant: \(15 = ab\)

### Step 3: Solve for \(b\)
From the first equation, \(a = 6\). Substituting this into the second equation:

\[
15 = 6b
\]

Now, solve for \(b\):

\[
b = \frac{15}{6} = 2.5
\]

Thus, the value of \(b\) is **2.5**.

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Would you like more details, or do you have any questions? Here are some related questions to expand on the concept:

1. What happens if the equation has no solution?
2. How do you determine if a system of equations has a unique solution?
3. How does changing the value of \(a\) affect the solution set?
4. What is the graphical interpretation of having infinitely many solutions?
5. How do you solve for \(b\) if the equation had only one solution?

**Tip:** Infinitely many solutions occur when both sides of an equation represent the same line in a system of linear equations.

In the equation 6x + 15 = a(x + b), if the equation has infinitely many solutions for x, what is the value of b?

Solve the equation 6x + 15 = a(x + b) for b when there are infinitely many solutions. This problem covers algebraic concepts, focusing on linear equations and the condition for infinitely many solutions.

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