- Identify two points from the table: \((2, 7)\) and \((3, 12)\).
- Apply the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) using these coordinates.
- Calculate the result: \(m = \frac{12 - 7}{3 - 2} = 5\).
- The slope of the line is \(\boxed{5}\).
Explanation
- Analyzing the Given Data and Understanding the Slope Formula
The problem asks for the slope of a line that passes through several given points. The points are provided in a table as: \((2, 7)\), \((3, 12)\), \((4, 17)\), and \((5, 22)\). A line can be described by the linear equation \(y = mx + b\), where \(m\) represents the slope and \(b\) is the \(y\)-intercept. The slope \(m\) measures the rate of change of \(y\) with respect to \(x\) and is calculated using any two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) on the line according to the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
- Calculating the Slope Using the Given Points
To find the slope \(m\), we select two distinct points from the table. Let us choose the first two points: \((x_1, y_1) = (2, 7)\) and \((x_2, y_2) = (3, 12)\).
Applying the slope formula:
\[m = \frac{12 - 7}{3 - 2}\]
Calculating the difference in the numerator and the denominator:
\[m = \frac{5}{1} = 5\]
Thus, the slope of the line is 5.
- Verifying the Slope and Final Conclusion
To ensure our result is consistent, let us verify the slope using two other points from the table, such as \((4, 17)\) and \((5, 22)\).
Applying the slope formula again:
\[m = \frac{22 - 17}{5 - 4}\]
Calculating the values:
\[m = \frac{5}{1} = 5\]
Since both calculations yield the same value, we can be confident that the slope of the line is indeed 5. The equation of the line is \(y = 5x - 3\). The slope is \(\boxed{5}\).
Examples
The concept of slope is fundamental in everyday life, particularly when dealing with rates of change. For example, if you are driving a car and observe the distance you travel over specific intervals of time, the "slope" of your distance-time graph represents your average speed. If you travel 7 miles in 2 hours, 12 miles in 3 hours, 17 miles in 4 hours, and 22 miles in 5 hours, the slope is 5 miles per hour, indicating a constant speed of 5 mph. This helps in predicting future positions and planning travel time efficiently.