Find the slope and the y-intercept of the line with the given equation.
$f(x)=7-\frac{4}{5} x$
The equation provided is \(f(x) = 7 - \frac{4}{5}x\). This is a linear function that describes a straight line when plotted on a Cartesian coordinate system. Our objective is to determine the slope and the y-intercept of this line. A linear equation is typically written in the slope-intercept form, given by \(f(x) = mx + b\), where \(m\) represents the slope (the rate of change) and \(b\) represents the y-intercept (the point where the line crosses the y-axis, i.e., the value of \(f(x)\) when \(x = 0\)).
To identify the slope and y-intercept, we rearrange the given equation \(f(x) = 7 - \frac{4}{5}x\) to match the slope-intercept form \(f(x) = mx + b\). By changing the order of the terms on the right side of the equation, we get:
\[f(x) = -\frac{4}{5}x + 7\]
Now, by comparing this directly to \(f(x) = mx + b\), we can clearly identify the values for \(m\) and \(b\). The coefficient of \(x\) is the slope, and the constant term is the y-intercept. Thus, the slope \(m\) is \(-\frac{4}{5}\) and the y-intercept \(b\) is \(7\).
In conclusion, for the linear function \(f(x) = 7 - \frac{4}{5}x\), the slope is \(-\frac{4}{5}\) and the y-intercept is \(7\). This means that for every unit increase in \(x\), the value of \(f(x)\) decreases by \(\frac{4}{5}\), and the line crosses the vertical axis at the height of \(7\). The slope is \(\boxed{-\frac{4}{5}}\) and the y-intercept is \(\boxed{7}\).
Linear functions are fundamental in modeling real-world scenarios with constant rates of change. For example, if you are draining a water tank that starts with 7 liters of water and loses \(\frac{4}{5}\) of a liter every minute, you can represent the remaining volume \(V\) after \(x\) minutes as \(V(x) = 7 - \frac{4}{5}x\). Knowing the slope helps you predict how fast the water level drops, and identifying the y-intercept tells you the initial amount of water in the tank before you began draining it.