a(x+b)=4x+10
In the equation above , a and b are constants. If the equation has infinitely many solutions for x , what is the value of b ?

Explanation

To solve the equation \( a(x + b) = 4x + 10 \) for the value of \( b \) when the equation has infinitely many solutions, we will follow these steps:

### Step 1: Expand the left-hand side
We start with the equation:
\[ a(x + b) = 4x + 10 \]
Expanding the left-hand side gives us:
\[ ax + ab = 4x + 10 \]

### Step 2: Compare coefficients
For the equation to have infinitely many solutions, the coefficients of \( x \) and the constant terms on both sides must be equal. Therefore, we can set up the following equations by comparing coefficients:
1. \( a = 4 \) (coefficient of \( x \))
2. \( ab = 10 \) (constant term)

### Step 3: Substitute \( a \) into the second equation
From the first equation, we have \( a = 4 \). We substitute this value into the second equation:
\[ 4b = 10 \]

### Step 4: Solve for \( b \)
Now, we solve for \( b \) by dividing both sides of the equation by 4:
\[ b = \frac{10}{4} = \frac{5}{2} \]

### Final Answer
Thus, the value of \( b \) is:
\[ \boxed{\frac{5}{2}} \]

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