Find the $y$-intercept of the line represented by the equation below.
$y=-\frac{1}{5} x$
The line is given by the equation \(y = -\frac{1}{5}x\). To find the \(y\)-intercept, we look for the point where the line crosses the \(y\)-axis. By definition, this occurs when the \(x\)-coordinate is equal to 0. We can substitute \(x = 0\) into the equation to find the corresponding \(y\)-value:
\[y = -\frac{1}{5}(0) = 0.\]
Thus, the \(y\)-intercept occurs at the point \((0, 0)\).
The standard slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the \(y\)-coordinate of the \(y\)-intercept. Comparing our given equation \(y = -\frac{1}{5}x\) with the standard form, we can write it as \(y = -\frac{1}{5}x + 0\). This makes it clear that the constant term \(b\) is 0. Therefore, the \(y\)-intercept of the line is 0, which corresponds to the point \((0, 0)\).
In real life, the y-intercept often represents a starting value in a linear relationship. For instance, if you are filling a water tank, the y-intercept could represent the amount of water already in the tank before you turn on the tap (time x=0). In this case, since the y-intercept is 0, it means the tank starts empty at the moment you begin recording the filling process. Understanding this helps in modeling various growth or decay patterns in economics, physics, and daily budgeting.