- Set \(y=0\) to find the \(x\)-intercept: \(x - 0 = 9 \implies x = 9\).
- Set \(x=0\) to find the \(y\)-intercept: \(0 - y = 9 \implies y = -9\).
- The \(x\)-intercept is \(\boxed{9}\) and the \(y\)-intercept is \(\boxed{-9}\).
Explanation
- Determining the Intercepts
To find the intercepts of the graph of the linear equation \(x - y = 9\), we need to identify the points where the line crosses the coordinate axes.
The \(x\)-intercept is the point on the graph where the line intersects the \(x\)-axis. At this point, the value of the \(y\)-coordinate is always 0. By setting \(y = 0\) in the given equation, we obtain \(x - 0 = 9\), which simplifies to \(x = 9\). Thus, the \(x\)-intercept is at the point \((9, 0)\), and the intercept value is 9.
The \(y\)-intercept is the point on the graph where the line intersects the \(y\)-axis. At this point, the value of the \(x\)-coordinate is always 0. By setting \(x = 0\) in the given equation, we obtain \(0 - y = 9\), which simplifies to \(-y = 9\). Solving for \(y\) by multiplying both sides by -1 gives \(y = -9\). Thus, the \(y\)-intercept is at the point \((0, -9)\), and the intercept value is -9.
Examples
Intercepts are fundamental in graphing linear equations, as they provide two easy-to-find points that define the entire line. For example, if you are tracking the fuel consumption of a car over time, the \(y\)-intercept could represent the starting amount of fuel in the tank, and the \(x\)-intercept could represent the time it takes until the tank is empty. Understanding these points helps in modeling and predicting trends in real-world scenarios.