The gradient of the line given by the equation $y = 4x - 1$ is $4$.

In the standard form of a linear equation, typically expressed as $y = mx + c$, the gradient (also known as the slope) is denoted by the coefficient $m$. This coefficient indicates the steepness of the line. In the equation $y = 4x - 1$, the coefficient of $x$ is $4$. Thus, we can conclude that the gradient of this line is $4$.

To clarify this concept further, let’s examine what the gradient represents. The gradient of a line quantifies how much the $y$-value changes in response to a one-unit change in the $x$-value. Specifically, a gradient of $4$ implies that for every $1$ unit increase in $x$, the $y$-value increases by $4$ units. This indicates that the line has a relatively steep incline.

To visualize this on a graph, consider starting at any point along the line. If you move $1$ unit to the right (along the $x$-axis), you would need to move $4$ units up (along the $y$-axis) to remain on the line. This consistent rate of change is what characterizes the gradient.

In summary, the gradient serves as a measure of both the steepness and the direction of a line. A positive gradient, such as $4$ in this example, indicates that the line slopes upward from left to right. Understanding gradients is essential for analyzing and interpreting linear relationships in various mathematical scenarios.