- To find the x-intercept, set \(y = 0\) in the equation \(6x + 7y = 18\) and solve for \(x\), which gives \(x = 3\).
- To find the y-intercept, set \(x = 0\) in the equation \(6x + 7y = 18\) and solve for \(y\), which gives \(y = \frac{18}{7}\).
- The x-intercept is \(\boxed{3}\).
- The y-intercept is \(\boxed{\frac{18}{7}}\).
Explanation
- Analysis of the Problem
To find the x-intercept and y-intercept of the line given by the equation \(6x + 7y = 18\), we need to identify the points where the graph intersects the axes. An x-intercept occurs where the graph crosses the x-axis, meaning the y-coordinate is 0. Similarly, a y-intercept occurs where the graph crosses the y-axis, meaning the x-coordinate is 0. We can determine these by substituting these values into the given linear equation.
- Calculating the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation \(6x + 7y = 18\). This simplifies to:
\[6x + 7(0) = 18\]
\[6x = 18\]
Dividing both sides by 6, we get:
\[x = \frac{18}{6} = 3\]
Thus, the x-intercept is 3.
- Calculating the y-intercept
To find the y-intercept, we set \(x = 0\) in the equation \(6x + 7y = 18\). This simplifies to:
\[6(0) + 7y = 18\]
\[7y = 18\]
Dividing both sides by 7, we get:
\[y = \frac{18}{7}\]
Thus, the y-intercept is \(\frac{18}{7}\).
- Conclusion
The x-intercept of the graph of \(6x + 7y = 18\) is 3, and the y-intercept is \(\frac{18}{7}\). Both are expressed in their simplest form. The x-intercept is \(\boxed{3}\) and the y-intercept is \(\boxed{\frac{18}{7}}\).
Examples
Understanding intercepts is essential for graphing linear equations quickly. In real life, this can represent finding the starting point or the end point of a process. For example, if a company's production cost is modeled by a linear equation, the y-intercept represents the fixed costs (when zero units are produced), while the x-intercept could represent the break-even point in terms of units produced if the equation represented revenue. Knowing these intercepts helps in quickly visualizing the behavior of such linear relationships.