- To find the x-intercept, set \(y = 0\) in the equation \(10x + y = 8\), resulting in \(x = \frac{4}{5}\).
- To find the y-intercept, set \(x = 0\) in the equation \(10x + y = 8\), resulting in \(y = 8\).
- The final answer is:
- x-intercept: \(\boxed{\frac{4}{5}}\)
- y-intercept: \(\boxed{8}\)
Explanation
- Understanding the Problem and Intercepts
The problem asks us to find the x-intercept and y-intercept of the line defined by the linear equation \(10x + y = 8\). An intercept represents the point where the line crosses one of the axes in the Cartesian plane. The x-intercept is the point where the line intersects the x-axis, which occurs when the y-coordinate is 0. Conversely, the y-intercept is the point where the line intersects the y-axis, which occurs when the x-coordinate is 0. We can find these values by substituting these coordinates into the given equation and solving for the remaining variable.
- Calculating the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation \(10x + y = 8\). This gives:
\[10x + 0 = 8\]
\[10x = 8\]
Dividing both sides by 10, we obtain:
\[x = \frac{8}{10}\]
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2, we get:
\[x = \frac{4}{5}\]
Thus, the x-intercept is \(\frac{4}{5}\).
- Calculating the y-intercept
To find the y-intercept, we set \(x = 0\) in the equation \(10x + y = 8\). This gives:
\[10(0) + y = 8\]
\[0 + y = 8\]
\[y = 8\]
Thus, the y-intercept is 8.
- Conclusion
We have determined the x-intercept and the y-intercept for the given linear equation. The x-intercept is \(\frac{4}{5}\) and the y-intercept is 8. The final results are \(x\)-intercept: \(\boxed{\frac{4}{5}}\) and \(y\)-intercept: \(\boxed{8}\).
Examples
Understanding how to find intercepts is a fundamental skill in graphing linear equations. For example, if you are analyzing a budget where \(10x + y = 8\) represents a relationship between two products you can buy (where \(x\) and \(y\) are quantities), the x-intercept tells you the maximum amount of product \(x\) you can purchase if you buy none of product \(y\), and the y-intercept tells you the maximum of product \(y\) if you buy none of product \(x\). This helps in identifying the boundaries of your choices on a coordinate plane.