Are there different ways to solve 3x3 matrices?

One approach avoids computing the determinant completely.

The determinant is zero if and only if the column vectors are linearly dependent. Since the left column and right column are independent, this means the middle column, $\begin{pmatrix}x\\4\\x\end{pmatrix}$ must be a linear combination of the left and right column.

This means there must be an $a,b$ such that: $$a+2b=x, 3a+5b=4, 6a+7b=x.$$ Then you have three equations in three variables, and it is easily solved, $a=-2,b=2,x=2.$

This works because all the occurrences of $x$ are in one column. The same would work if all occurrences of $x$ were in a row.