- To find the \(x\)-intercept, set \(y = 0\) in the equation \(-2x + 4y = 40\), which results in \(-2x = 40\), so \(x = -20\).
- To find the \(y\)-intercept, set \(x = 0\) in the equation \(-2x + 4y = 40\), which results in \(4y = 40\), so \(y = 10\).
- The \(x\)-intercept is \(\boxed{-20}\) and the \(y\)-intercept is \(\boxed{10}\).
Explanation
- Understanding Intercepts
To find the intercepts of the graph of the linear equation \(-2x + 4y = 40\), we look for the points where the graph crosses the coordinate axes. The \(x\)-intercept occurs where the graph crosses the \(x\)-axis, which means the \(y\)-coordinate is zero. The \(y\)-intercept occurs where the graph crosses the \(y\)-axis, which means the \(x\)-coordinate is zero. We will solve the equation for these two conditions.
- Calculating the x-intercept
To find the \(x\)-intercept, we set \(y = 0\) in the equation \(-2x + 4y = 40\):
\[-2x + 4(0) = 40\]
\[-2x = 40\]
Dividing both sides by \(-2\), we get:
\[x = \frac{40}{-2} = -20\]
Thus, the \(x\)-intercept is \(-20\).
- Calculating the y-intercept
To find the \(y\)-intercept, we set \(x = 0\) in the equation \(-2x + 4y = 40\):
\[-2(0) + 4y = 40\]
\[4y = 40\]
Dividing both sides by \(4\), we get:
\[y = \frac{40}{4} = 10\]
Thus, the \(y\)-intercept is \(10\).
- Conclusion
We have determined that the \(x\)-intercept is \(-20\) and the \(y\)-intercept is \(10\). These values indicate that the line crosses the \(x\)-axis at \((-20, 0)\) and the \(y\)-axis at \((0, 10)\). The final answers are \(\boxed{-20}\) and \(\boxed{10}\).
Examples
The concept of intercepts is essential in real-world applications such as economics or engineering. For example, in budgeting, if a company has a fixed amount of revenue (\(\$40\)) and two products to produce with different costs per unit (\(\$2\) and \(\$4\)), the intercept points show the maximum amount of one product they can produce if they produce none of the other. Finding these limits helps in understanding the boundaries of resource allocation. The equation \(-2x + 4y = 40\) acts as a linear model for balancing such trade-offs.