Find the slope of the line determined by the following equation. Simplify your answer.
$-8 y=7$
The given equation is \(-8y = 7\). We need to determine the slope of the line represented by this equation. A linear equation in two variables can often be written in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Alternatively, we can analyze the geometric properties of the equation. In this case, the equation contains only the variable \(y\), which implies that the value of \(y\) is constant regardless of the value of \(x\).
To find the slope, we first solve for \(y\) by dividing both sides of the equation by \(-8\):
\[-8y = 7\]
\[y = -\frac{7}{8}\]
This equation defines a horizontal line where for every value of \(x\), the \(y\)-coordinate is always \(-\frac{7}{8}\). Any equation of the form \(y = k\), where \(k\) is a constant, represents a horizontal line. A horizontal line has a constant \(y\)-value, meaning there is no "rise" as we move along the \(x\)-axis. Therefore, the change in \(y\) is always \(0\) for any change in \(x\). The slope \(m\) is defined as the change in \(y\) divided by the change in \(x\), which is \(\frac{0}{\Delta x} = 0\).
Since the line is horizontal, its slope is \(\boxed{0}\).
The concept of slope is crucial in engineering and physics, as it represents the rate of change of one variable with respect to another. For example, if you are designing a perfectly level shelf, the "slope" of the surface relative to the ground is 0. This ensures that objects placed on the shelf do not slide off due to gravity. Understanding that horizontal lines have a slope of 0 allows us to model stationary or level conditions in various real-world systems.