Answer: B. $-7$ only

Explanation:1. Understand the condition for infinitely many solutions: An equation of the form $Ax + B = Cx + D$ has infinitely many solutions if and only if $A=C$ and $B=D$. This means the coefficients of x on both sides must be equal, and the constant terms on both sides must be equal, resulting in an identity like $0x + k = k$ or simply $k=k$.

2. Expand and simplify both sides of the given equation: Left side: $a(7x-14)+5a = 7ax - 14a + 5a = 7ax - 9a$

Right side: $7(ax-2a)-35 = 7ax - 14a - 35$

3. Set the simplified sides equal: $7ax - 9a = 7ax - 14a - 35$

4. Apply the condition for infinitely many solutions: For this equation to have infinitely many solutions, the coefficients of x on both sides must be equal, and the constant terms on both sides must be equal. Coefficient of x: The coefficient of x on the left side is $7a$. The coefficient of x on the right side is $7a$. Setting them equal: $7a = 7a$. This is true for any value of a, and doesn't help us find a specific value for 'a' yet.

Constant terms: The constant term on the left side is $-9a$. The constant term on the right side is $-14a - 35$. Setting them equal: $-9a = -14a - 35$

5. Solve the equation for the constant terms to find 'a': $-9a = -14a - 35$ Add $14a$ to both sides: $-9a + 14a = -35$ $5a = -35$ Divide by $5$: $a = \dfrac{-35}{5}$ $a = -7$

6. Check the solution with the options: The value $a = -7$ makes both the coefficients of x equal (which they always are here) AND the constant terms equal. Let's verify: If $a=-7$, the equation becomes $7(-7)x - 9(-7) = 7(-7)x - 14(-7) - 35$ $-49x + 63 = -49x + 98 - 35$ $-49x + 63 = -49x + 63$ This is an identity, so it has infinitely many solutions when $a=-7$. Option [B] is $-7$ only.

7. Conclusion: For the equation to have infinitely many solutions, the value of a must be $-7$.