- Identify the formula for the slope of a line passing through two points: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- Select two points from the table, \((-24, 0)\) and \((-9, 3)\), and substitute them into the formula: \(m = \frac{3 - 0}{-9 - (-24)} = \frac{3}{15}\).
- Simplify the fraction to obtain the slope: \(m = \frac{1}{5}\).
- Verify the consistency of the slope using another pair of points, \((6, 6)\) and \((21, 9)\), which confirms that \(m = \frac{9 - 6}{21 - 6} = \frac{3}{15} = \frac{1}{5}\).
- State the final answer: \(\boxed{\frac{1}{5}}\).
Explanation
- Analysis of the Problem and Slope Definition
The objective is to determine the slope of a line that passes through the set of points provided in the table: \((-24, 0), (-9, 3), (6, 6),\) and \((21, 9).\) The slope, often denoted by \(m\), represents the rate of change of the variable \(y\) with respect to the variable \(x\). For any two distinct points \((x_1, y_1)\) and \((x_2, y_2)\) on a straight line, the slope is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Because the given points lie on a single straight line, the ratio of the change in \(y\) to the change in \(x\) will be constant for any pair of these points.
- Calculating the Slope Using Two Points
To calculate the slope, we select two distinct points from the given table, for instance, \((-24, 0)\) and \((-9, 3).\) Substituting these coordinates into the slope formula:
\[x_1 = -24, y_1 = 0\]
\[x_2 = -9, y_2 = 3\]
\[m = \frac{3 - 0}{-9 - (-24)}\]
Simplifying the expression:
\[m = \frac{3}{-9 + 24} = \frac{3}{15}\]
Reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3, we obtain:
\[m = \frac{3 \div 3}{15 \div 3} = \frac{1}{5}\]
Thus, the slope calculated from these two points is \(\frac{1}{5}\).
- Verification and Final Conclusion
To verify that the slope is consistent, we can choose another pair of points from the table, such as \((6, 6)\) and \((21, 9).\) Using the slope formula again:
\[x_1 = 6, y_1 = 6\]
\[x_2 = 21, y_2 = 9\]
\[m = \frac{9 - 6}{21 - 6}\]
Simplifying this expression:
\[m = \frac{3}{15}\]
Reducing the fraction again:
\[m = \frac{1}{5}\]
Since we obtained the same result, we have confirmed that the slope of the line passing through all these points is \(\frac{1}{5}\). The final answer is \(\boxed{\frac{1}{5}}\).
Examples
The concept of slope is vital in everyday life, for example, in construction or engineering. Imagine you are building a wheelchair ramp; the steepness of the ramp is defined by its slope, which is the ratio of its vertical height (rise) to its horizontal length (run). By calculating the slope as \(\frac{\text{rise}}{\text{run}}\), engineers ensure that the ramp is safe and accessible for everyone. A slope like \(\frac{1}{5}\) means that for every 5 units of horizontal distance, the ramp rises 1 unit vertically, maintaining a gentle and manageable incline.