What is the slope of the line passing through the points $(3,5)$ and $(7,13)$ ?
The objective is to find the slope of the line passing through two points: \((x_1, y_1) = (3, 5)\) and \((x_2, y_2) = (7, 13)\). The slope \(m\) of a line connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is defined as the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate, which is expressed by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This value represents the constant rate of change of the line, indicating how much the \(y\)-value increases or decreases for every unit increase in \(x\).
To find the slope, we substitute the coordinates of our given points \((3, 5)\) and \((7, 13)\) into the slope formula:
\[m = \frac{13 - 5}{7 - 3}\]
First, calculate the numerator (the difference in \(y\)-coordinates):
\[13 - 5 = 8\]
Next, calculate the denominator (the difference in \(x\)-coordinates):
\[7 - 3 = 4\]
Now, perform the division:
\[m = \frac{8}{4} = 2\]
Thus, the slope of the line passing through the points \((3, 5)\) and \((7, 13)\) is \(2\). Comparing this result to the given options (A: \(1/2\), B: \(2\), C: \(-2\), D: \(-1/2\)), we identify that the correct option is B.
In real life, the concept of slope is used to describe steepness. For example, if you are designing a wheelchair ramp, the slope helps you determine how high the ramp rises for every horizontal distance covered, ensuring the ramp is safe for use. By knowing the starting point \((x_1, y_1)\) and the ending point \((x_2, y_2)\), you can calculate the slope to ensure it complies with accessibility guidelines. In our specific case with points \((3,5)\) and \((7,13)\), the steepness is 2.