Let me first say that the solution in the textbook is wrong.

So if you are wondering whether your solution is correct or not, maybe your is correct.

Or do you want to know how to solve the problem?

I would indeed like to know how to solve this problem. I got an answer but I am not sure that it is correct or that I even used the correct method to solve it.

Hint.

1. Write the augmented matrix and reduce it.

2. If the homogeneous system has a nonzero solution, then it has at least one free variable.

3. The number of free variables is $n-r$, where $n$ is the number of unknowns and $r$ is rank.

4. So, rank must be less than $3$.

5. Find the value of a so that the rank is $2$.