- The \(x\)-intercept is found by setting \(y = 0\), which gives \(x = 34\).
- The \(y\)-intercept is found by setting \(x = 0\), leading to \(7y = 34\), so \(y = \frac{34}{7}\).
- The intercepts are \(\boxed{34}\) and \(\boxed{\frac{34}{7}}\).
Explanation
- Understanding the Problem
The equation provided is a linear equation in two variables, \(x + 7y = 34\). A linear equation represents a straight line in the coordinate plane. Intercepts are specific points where this line crosses the coordinate axes: the \(x\)-intercept is where the line crosses the \(x\)-axis (at which point \(y = 0\)), and the \(y\)-intercept is where the line crosses the \(y\)-axis (at which point \(x = 0\)). Finding these intercepts allows us to quickly identify key points on the line and visualize its position.
- Finding the x-intercept
To find the \(x\)-intercept, we set \(y = 0\) in the given equation and solve for \(x\):
\[x + 7(0) = 34\]
\[x + 0 = 34\]
\[x = 34\]
Thus, the \(x\)-intercept is \(34\), which corresponds to the coordinate point \((34, 0)\).
- Finding the y-intercept
To find the \(y\)-intercept, we set \(x = 0\) in the given equation and solve for \(y\):
\[0 + 7y = 34\]
\[7y = 34\]
Dividing both sides by 7, we obtain:
\[y = \frac{34}{7}\]
The \(y\)-intercept is the improper fraction \(\frac{34}{7}\), which corresponds to the coordinate point \((0, \frac{34}{7})\).
Examples
Understanding intercepts is crucial for graphing linear equations, which is a fundamental skill in algebra. For instance, if you are tracking the distance of a train from a station over time, the \(y\)-intercept represents the initial distance when time is zero, and the \(x\)-intercept can represent the time at which the train arrives at the station (distance is zero). Calculating these points makes it easy to plot the movement of the object on a graph and analyze its behavior.