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The slope of a line measures how steep the line is, which can indicate whether it rises, falls, or remains flat as it moves from left to right. Mathematically, the slope is often represented by the letter \(m\). With two known points on a line, say \((x_1, y_1)\) and \((x_2, y_2)\), you can calculate the slope using:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula calculates how much the y-values (vertical change) change per unit of x-value (horizontal change).

In our example, we have the points \((-1,4)\) and \((6,4)\). Plugging these into the slope formula we find:

• \(m = \frac{4 - 4}{6 - (-1)} = \frac{0}{7} = 0\)

This zero result indicates a perfectly horizontal line, as there is no vertical change between the points.

The slope-intercept form of a straight line is a powerful way to describe the line’s mathematical relationship. It is written as:

• \(y = mx + b\)

In this equation, \(m\) is the slope, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

To find the equation for the line through our points \((-1,4)\) and \((6,4)\), we substitute the slope \(m = 0\) into the slope-intercept form. Since the line is horizontal, this results in:

• \(y = 0x + b\) simplifies to \(y = b\)

Using one of the points, such as \((-1, 4)\), we substitute into the equation to find \(b\). Thus, \(4 = 0(-1) + b\), giving us \(b = 4\). Therefore, the line's equation is simply \(y = 4\).

A horizontal line is a unique type of line where every point has the same y-coordinate. As a result, its slope is always 0, reflecting no vertical change, regardless of the x-coordinate.

This absence of slope leads to a straightforward equation in the form \(y = b\), meaning the line doesn’t tilt up or down but rather extends equally in each horizontal direction. For the line in our task, the equation \(y = 4\) represents every point along the line sharing the y-coordinate of 4, like \((-1, 4)\) and \((6, 4)\).

In graphical terms, a horizontal line is parallel to the x-axis and crosses the y-axis at \(b\), which is the y-intercept of the line. Visualizing it on a cartesian plane, it runs straight across, creating an easy-to-spot line for any given y-value.