What is the $x$-intercept of the line with this equation $\frac{1}{3} x+y=-15$?
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To find the \(x\)-intercept of a line given by the equation \(\frac{1}{3}x + y = -15\), we must determine the point where the line crosses the \(x\)-axis. By definition, at any point on the \(x\)-axis, the \(y\)-coordinate is equal to \(0\). Therefore, we need to find the value of \(x\) that satisfies the equation when \(y = 0\).
We substitute \(y = 0\) into the given equation:
\[\frac{1}{3}x + (0) = -15\]
This simplifies to:
\[\frac{1}{3}x = -15\]
To isolate \(x\), we multiply both sides of the equation by \(3\):
\[x = -15 \times 3\]
Performing the multiplication, we find:
\[x = -45\]
Since we found that \(x = -45\) when \(y = 0\), the \(x\)-intercept of the line is the point \((-45, 0)\). The value we were looking for is \(-45\), meaning the coordinate of the \(x\)-intercept is \(\boxed{(-45, 0)}\).
The concept of finding an intercept is highly useful in physics and economics. For example, if you are tracking the fuel level in a tank over time, represented by a linear equation, the x-intercept would represent the specific time at which the fuel tank becomes completely empty (where the fuel amount y equals 0). This allows for efficient planning of refueling stops and logistics management based on fuel consumption rates.