To determine the number of solutions to the equation
[tex]\[ 6x + 35 + 9x = 15(x + 4) - 25, \][/tex]
we'll follow the step-by-step process of simplifying and analyzing the equation.
### Step 1: Simplify both sides of the equation
First, combine like terms on each side of the equation starting with the left-hand side:
[tex]\[ 6x + 9x + 35 = 15x \][/tex]
This simplifies to:
[tex]\[ 15x + 35 \][/tex]
Next, expand on the right-hand side:
[tex]\[ 15(x + 4) - 25 \][/tex]
[tex]\[ 15x + 60 - 25 \][/tex]
Simplify the right-hand side:
[tex]\[ 15x + 35 \][/tex]
### Step 2: Set the simplified forms equal to each other
Now we equate the simplified left-hand side to the simplified right-hand side:
[tex]\[ 15x + 35 = 15x + 35 \][/tex]
### Step 3: Analyze the resulting equation
Notice that the equation:
[tex]\[ 15x + 35 = 15x + 35 \][/tex]
is true for all values of [tex]\( x \)[/tex]. This means that no matter what value [tex]\( x \)[/tex] takes, the equation will always hold true.
### Step 4: Conclusion
Since the equation is valid for all values of [tex]\( x \)[/tex], there are infinitely many solutions.
The number of solutions to the equation is:
[tex]\[ \text{B. Infinitely many} \][/tex]
