A line that includes the points $(d, 9)$ and $(-5,6)$ has a slope of $\frac{1}{5}$. What is the value of $d$?
We are given a line that passes through two points, \(P_1 = (d, 9)\) and \(P_2 = (-5, 6)\), and has a slope \(m = \frac{1}{5}\). The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is determined by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Our task is to use this information to determine the value of the unknown coordinate \(d\).
Substituting the given points and the slope into the slope formula, we have:
\[\frac{1}{5} = \frac{6 - 9}{-5 - d}\]
We simplify the numerator on the right side:
\[6 - 9 = -3\]
So the equation becomes:
\[\frac{1}{5} = \frac{-3}{-5 - d}\]
To simplify the equation, we can write the right-hand side by multiplying the numerator and denominator by \(-1\):
\[\frac{-3}{-5 - d} = \frac{3}{5 + d}\]
Now the equation is:
\[\frac{1}{5} = \frac{3}{5 + d}\]
We can solve for \(d\) by cross-multiplying:
\[1 \times (5 + d) = 5 \times 3\]
\[5 + d = 15\]
Subtracting 5 from both sides:
\[d = 15 - 5 = 10\]
Thus, the value of \(d\) is \(10\).
In architecture and engineering, determining the slope of a ramp or a roof is essential for both aesthetic design and structural integrity. For example, if you are designing a wheelchair ramp that needs to rise a certain height (the y-coordinate difference) over a specific horizontal distance (the x-coordinate difference), knowing the slope allows you to calculate the required base placement point \(d\). Understanding how to relate coordinates, slopes, and linear equations is a fundamental skill in these practical design tasks.