- The x-intercept is found by setting \(y = 0\) in \(8x - 10y = 32\), which yields \(x = 4\).
- The y-intercept is found by setting \(x = 0\) in \(8x - 10y = 32\), which yields \(y = -\frac{16}{5}\).
- The intercepts are \(\boxed{4}\) for \(x\) and \(\boxed{-\frac{16}{5}}\) for \(y\).
Explanation
- Understanding the Problem
To find the intercepts of the graph defined by the equation \(8x - 10y = 32\), we look for the points where the graph crosses the coordinate axes. The x-intercept occurs where the y-coordinate is 0, and the y-intercept occurs where the x-coordinate is 0. Both intercepts provide key information about the position and orientation of the line on a coordinate plane.
- Finding the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation \(8x - 10y = 32\):
\[8x - 10(0) = 32\]
\[8x = 32\]
Dividing both sides by 8, we get:
\[x = \frac{32}{8} = 4\]
Thus, the x-intercept is \(4\).
- Finding the y-intercept
To find the y-intercept, we set \(x = 0\) in the equation \(8x - 10y = 32\):
\[8(0) - 10y = 32\]
\[-10y = 32\]
Dividing both sides by -10, we get:
\[y = \frac{32}{-10}\]
Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, 2:
\[y = -\frac{16}{5}\]
Thus, the y-intercept is \(-\frac{16}{5}\).
- Conclusion
In summary, the x-intercept of the line is \(4\) and the y-intercept is \(-\frac{16}{5}\). These values correspond to the points \((4, 0)\) and \((0, -\frac{16}{5})\) respectively, which characterize the linear relationship between \(x\) and \(y\). Thus, the x-intercept is \(\boxed{4}\) and the y-intercept is \(\boxed{-\frac{16}{5}}\).
Examples
The concept of intercepts is essential in fields like physics and economics. For example, if a company's production cost is modeled by a linear equation where \(x\) is the number of units and \(y\) is the total cost, the y-intercept represents the fixed costs (when no units are produced), while the x-intercept could represent the break-even point in terms of units. Calculating these points allows professionals to understand the starting conditions and critical thresholds of their systems.