The equation \( y = 4x + 1 \) represents a linear equation in two variables (x and y). Since this is a straight line, it has an infinite number of solutions. For any given value of \( x \), you can find a corresponding value of \( y \) by plugging \( x \) into the equation. Conversely, for any value of \( y \), you can find \( x \) by rearranging the equation.
In summary, the equation \( y = 4x + 1 \) has infinitely many solutions.