$y=-\frac{1}{3} x+1$
The equation provided is \(y = -\frac{1}{3}x + 1\). This is a linear equation representing a straight line in the Cartesian coordinate plane. It is written in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. In this specific case, we identify the slope as \(m = -\frac{1}{3}\) and the y-intercept as \(b = 1\), which corresponds to the point \((0, 1)\).
To find the x-intercept, we set the value of \(y\) to \(0\) in the equation. This gives us:
\[0 = -\frac{1}{3}x + 1\]
Adding \(\frac{1}{3}x\) to both sides of the equation yields:
\[\frac{1}{3}x = 1\]
Multiplying both sides by \(3\) results in:
\[x = 3\]
Thus, the x-intercept of the line is the point \((3, 0)\).
The slope of the line is \(m = -\frac{1}{3}\). Because the slope is negative (\(m < 0\)), the line is decreasing, meaning that as the value of \(x\) increases, the value of \(y\) decreases. The line passes through the y-intercept \((0, 1)\) and the x-intercept \((3, 0)\). These two points define the unique straight line represented by the equation.
In real-life scenarios, linear equations like \(y = -\frac{1}{3}x + 1\) can model various relationships where one quantity changes at a constant rate relative to another. For instance, imagine you have a budget of \(\$1\) to spend on a service that charges a small fee per unit of time, where you lose \(\$ \frac{1}{3}\) of your remaining budget for every unit increase in time. If \(y\) represents your remaining budget and \(x\) represents the units of time passed, this equation tracks how your budget decreases over time until it runs out completely at \(x=3\). Understanding this helps in managing resources efficiently.