91Ó°ÊÓ

In algebra, the slope of a line is an important concept that indicates its steepness or inclination. When we talk about the slope, we often refer to it as "rise over run," which means how much the line rises vertically for each unit it runs horizontally.
Given two points, such as \((-1, 4)\) and \((6, 4)\), the formula for finding the slope \(m\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Plugging in the points \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (6, 4)\), we determine that:

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

A slope of zero tells us that the line is perfectly horizontal, meaning there is no vertical change as you move along the line. Understanding the slope is crucial for analyzing and graphing linear equations since it helps define the direction and behavior of the line.

The y-intercept of a line is the point where the line intersects the y-axis. This is where \(x\) is equal to zero. In the equation of a line given by \(y = mx + b\), the value \(b\) represents the y-intercept.
To find the y-intercept, we utilize the known equation \(y = mx + b\) and one of the points given, along with the slope \(m\).
Using point \((-1, 4)\) and a slope of \(0\):

• \(4 = 0 \times (-1) + b\)
• Therefore, \(b = 4\)

The y-intercept is \(4\), which means the line crosses the y-axis at \( (0, 4) \). Understanding the y-intercept allows you to easily locate where the line begins on the graph relative to the y-axis, thus serving as a pivotal anchor point for graphing a line.

Graphing a linear equation involves plotting points and understanding its path on the coordinate plane. In this case, our line equation is \(y = 4\). This specific line is horizontal, due to its slope being zero. To start plotting the points:

• Begin at the y-intercept point \((0, 4)\)
• Plot the given points, like \((-1, 4)\) and \((6, 4)\)

As you plot more points with the same y-value on the graph, you'll see that all these points line up horizontally at \(y=4\). The graph of \(y=4\) is a straight horizontal line running parallel to the x-axis and crossing the y-axis at \( (0, 4) \). Graphing becomes a straightforward task with a clear understanding of the slope and y-intercept, giving you a complete picture of how the linear equation behaves on the plane.