Perform the matrix multiplication on the left side of the equation.

The product of a $2 \times 2$ matrix and a $2 \times 1$ matrix is a $2 \times 1$ matrix.

$$\begin{pmatrix} 2 & x \\ 5 & 1 \end{pmatrix} \begin{pmatrix} 4 \\ 3 \end{pmatrix} = \begin{pmatrix} (2)(4) + (x)(3) \\ (5)(4) + (1)(3) \end{pmatrix}$$ $$= \begin{pmatrix} 8 + 3x \\ 20 + 3 \end{pmatrix}$$ $$= \begin{pmatrix} 8 + 3x \\ 23 \end{pmatrix}$$

From the first row:

$$8 + 3x = 17$$

From the second row:

$$23 = 17$$

The second equation $23 = 17$ is a contradiction, which indicates there might be an error in the problem statement or the given matrices. However, assuming the problem intends for us to find $x$ based on the first row's equality, we proceed with solving for $x$.

$$8 + 3x = 17$$

Subtract 8 from both sides:

$$3x = 17 - 8$$ $$3x = 9$$

Divide by 3:

$$x = \frac{9}{3}$$ $$x = 3$$

The value of $x$ is $\boxed{3}$.