A line has a slope of $-\frac{1}{8}$ and includes the points $(2,0)$ and $(p,-1)$. What is the value of $p$?
The problem asks us to find the value of \(p\) for a line that has a known slope of \(-\frac{1}{8}\) and passes through the two points \((2, 0)\) and \((p, -1)\). The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]
We have \(m = -\frac{1}{8}\), \((x_1, y_1) = (2, 0)\), and \((x_2, y_2) = (p, -1)\). Our objective is to set up an equation using these values and solve for \(p\).
Using the points \((2, 0)\) and \((p, -1)\), we substitute these into the slope formula:
\[-\frac{1}{8} = \frac{-1 - 0}{p - 2}.\]
This simplifies to:
\[-\frac{1}{8} = \frac{-1}{p - 2}.\]
To solve for \(p\), we can multiply both sides of the equation by \(-1\):
\[\frac{1}{8} = \frac{1}{p - 2}.\]
By comparing the denominators, we see that \(p - 2 = 8\). Adding 2 to both sides gives \(p = 10\).
We found the value of \(p\) to be 10. We can verify this by checking the slope with the points \((2, 0)\) and \((10, -1)\):
\[m = \frac{-1 - 0}{10 - 2} = \frac{-1}{8} = -\frac{1}{8}.\]
The calculated slope matches the given slope, confirming our result. Therefore, the final value is \(p = 10\).
In real life, the concept of slope is used to measure the steepness of objects like roofs or ramps. If an architect designs a handicap-accessible ramp with a specific slope, they need to ensure the vertical rise matches the horizontal run according to their measurements. For instance, if the plan specifies a certain rise for a given length of the ramp, they use the slope formula to calculate the necessary coordinates, ensuring the ramp is safe and meets building codes.