What is the slope of the line that passes through the points (3, 8) and (-2, 13)? Write your answer in simplest form.
We are given two points on a Cartesian plane: \((x_1, y_1) = (3, 8)\) and \((x_2, y_2) = (-2, 13)\). The objective is to find the slope of the line that passes through these two points. The slope \(m\) of a line indicates its steepness and direction, and it is calculated as the change in the \(y\)-coordinates (rise) divided by the change in the \(x\)-coordinates (run).
The formula to calculate the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Using our given points \((3, 8)\) and \((-2, 13)\), we set \(x_1 = 3\), \(y_1 = 8\), \(x_2 = -2\), and \(y_2 = 13\).
Now we substitute the values into the formula:
\[m = \frac{13 - 8}{-2 - 3}\]
First, calculate the numerator (the vertical change): \(13 - 8 = 5\).
Next, calculate the denominator (the horizontal change): \(-2 - 3 = -5\).
The expression becomes:
\[m = \frac{5}{-5}\]
Finally, simplify the fraction by dividing the numerator by the denominator:
\[m = -1\]
Thus, the slope of the line passing through the points \((3, 8)\) and \((-2, 13)\) is \(-1\).
In civil engineering and architecture, the slope of a line, often called the "grade" or "pitch," is critical. For example, when designing a wheelchair ramp, the slope must be carefully controlled to ensure safety and accessibility. By understanding the coordinates of the start and end points of a ramp path, you can use the slope formula to ensure that the rise of the ramp relative to its horizontal distance meets specific building codes or safety standards.