A line has a slope of $-\frac{1}{7}$ and includes the points $(6, u)$ and $(-1,-7)$. What is the value of $u$?
$u=$
The problem asks us to find the unknown coordinate \(u\) given a line with a specified slope \(m = -\frac{1}{7}\) that passes through two points: \(P_1 = (6, u)\) and \(P_2 = (-1, -7)\). We recall the definition of the slope \(m\) for a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]
This formula describes the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. We will use this to set up an equation for \(u\).
We substitute the coordinates of our two points, \(P_1 = (6, u)\) and \(P_2 = (-1, -7)\), into the slope formula, using \(m = -\frac{1}{7}\):
\[-\frac{1}{7} = \frac{-7 - u}{-1 - 6}.\]
First, we simplify the denominator on the right side:
\[-1 - 6 = -7.\]
Substituting this back into our equation, we have:
\[-\frac{1}{7} = \frac{-7 - u}{-7}.\]
We can now solve for \(u\) by simplifying the equation. Multiplying both sides by \(-7\), we get:
\[-\frac{1}{7} \times (-7) = \frac{-7 - u}{-7} \times (-7).\]
This simplifies to:
\[1 = -7 - u.\]
Now, we isolate \(u\) by adding \(u\) to both sides and subtracting 1 from both sides:
\[u = -7 - 1.\]
\[u = -8.\]
We have found that \(u = -8\). To verify, we substitute this back into the points to ensure they yield the given slope:
Points are \((6, -8)\) and \((-1, -7)\).
\[m = \frac{-7 - (-8)}{-1 - 6} = \frac{-7 + 8}{-7} = \frac{1}{-7} = -\frac{1}{7}.\]
Since the calculated slope matches the given slope, our result is correct. The value of \(u\) is \(-8\).
In navigation, a pilot might need to maintain a specific glide slope while descending to an airport. If the aircraft is at a specific horizontal distance and altitude, and the required glide slope is known, the pilot can use these coordinates to determine the necessary altitude adjustment at a different point in the flight path to maintain the consistent descent. Understanding the relationship between coordinates and slopes helps in maintaining precision in various engineering and logistics tasks.