The points $(w, 1)$ and $(0,5)$ fall on a line with a slope of $-\frac{4}{7}$. What is the value of $w$?
We are given two points on a line, \(P_1 = (w, 1)\) and \(P_2 = (0, 5)\), and the slope of this line is \(m = -\frac{4}{7}\). Our objective is to determine the value of the unknown coordinate \(w\). The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is determined by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula will allow us to relate the given points and the slope in a single linear equation to solve for \(w\).
Using the coordinates \(P_1 = (w, 1)\) as \((x_1, y_1)\) and \(P_2 = (0, 5)\) as \((x_2, y_2)\), we substitute these into the slope formula:
\[-\frac{4}{7} = \frac{5 - 1}{0 - w}\]
Simplifying the numerator on the right side, we get:
\[-\frac{4}{7} = \frac{4}{-w}\]
To solve for \(w\), we can use cross-multiplication:
\[-4 \times (-w) = 7 \times 4\]
\[4w = 28\]
Dividing both sides by 4, we find:
\[w = \frac{28}{4} = 7\]
Thus, the value of \(w\) is 7.
We have determined that \(w = 7\). Verifying this, the slope between \((7, 1)\) and \((0, 5)\) is \(\frac{5 - 1}{0 - 7} = \frac{4}{-7} = -\frac{4}{7}\), which matches the given information. The value of \(w\) is \(\boxed{7}\).
The concept of slope is fundamental in many real-world scenarios, such as designing a wheelchair ramp or determining the grade of a road. For instance, if a road needs to climb 5 meters for every 7 meters of horizontal distance, engineers use the slope to ensure safety and accessibility. By understanding how the slope relates coordinates on a plane, we can model physical structures, analyze trends in data, or even predict future outcomes based on linear growth or decline.