- To find the x-intercept, set \(y=0\) in the equation \(-2x + 4y = 40\), which simplifies to \(-2x = 40\), yielding \(x = -20\).
- To find the y-intercept, set \(x=0\) in the equation \(-2x + 4y = 40\), which simplifies to \(4y = 40\), yielding \(y = 10\).
- The x-intercept is \(\boxed{-20}\).
- The y-intercept is \(\boxed{10}\).
Explanation
- Understanding the Intercepts
The goal is to find the x-intercept and y-intercept of the linear equation \(-2x + 4y = 40\). An x-intercept is the point where the graph crosses the x-axis, occurring where \(y = 0\). A y-intercept is the point where the graph crosses the y-axis, occurring where \(x = 0\). We will solve for each variable by setting the other to zero.
- Calculating the x-intercept
To find the x-intercept, we substitute \(y = 0\) into the equation \(-2x + 4y = 40\):
\[-2x + 4(0) = 40\]
\[-2x = 40\]
Dividing both sides by \(-2\), we get:
\[x = -20\]
Thus, the x-intercept occurs at the point \((-20, 0)\), and the x-intercept value is \(-20\).
- Calculating the y-intercept
To find the y-intercept, we substitute \(x = 0\) into the equation \(-2x + 4y = 40\):
\[-2(0) + 4y = 40\]
\[4y = 40\]
Dividing both sides by \(4\), we get:
\[y = 10\]
Thus, the y-intercept occurs at the point \((0, 10)\), and the y-intercept value is \(10\).
- Conclusion
We have successfully determined the intercepts by setting one coordinate to zero and solving for the other. The x-intercept is \(-20\) and the y-intercept is \(10\).
\(x\)-intercept: \(\boxed{-20}\)
\(y\)-intercept: \(\boxed{10}\)
Examples
In real-world applications, finding intercepts is essential for analyzing trends in linear relationships. For example, if a company's profit \(P\) over time \(t\) follows a linear model like \(P = mt + b\), the y-intercept (\(t=0\)) represents the initial investment or starting state, while the t-intercept (\(P=0\)) represents the "break-even point," the exact time when the company stops losing money and starts generating profit. Identifying these points allows businesses to make informed decisions about their financial trajectory.