Question
Find and interpret the slope of the line containing the given points. (7,−1) and (4,3)
Answers
GPT-4o mini
To find the slope of the line containing the points \((7, -1)\) and \((4, 3)\), we use the formula for the slope \(m\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, we can assign:
- \((x_1, y_1) = (7, -1)\)
- \((x_2, y_2) = (4, 3)\)
Now, substituting these values into the slope formula:
\[
m = \frac{3 - (-1)}{4 - 7}
\]
This simplifies to:
\[
m = \frac{3 + 1}{4 - 7} = \frac{4}{-3} = -\frac{4}{3}
\]
The slope of the line is \(-\frac{4}{3}\).
### Interpretation of the Slope:
The slope \(-\frac{4}{3}\) indicates a negative relationship between the x-values and y-values of the points. Specifically, for every increase of 3 units in the \(x\) direction, the \(y\) value decreases by 4 units. In practical terms, this means that as you move to the right along the x-axis, the line goes down at a rate of 4 units down for every 3 units you move right. This signifies a downward slant of the line from left to right.
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, we can assign:
- \((x_1, y_1) = (7, -1)\)
- \((x_2, y_2) = (4, 3)\)
Now, substituting these values into the slope formula:
\[
m = \frac{3 - (-1)}{4 - 7}
\]
This simplifies to:
\[
m = \frac{3 + 1}{4 - 7} = \frac{4}{-3} = -\frac{4}{3}
\]
The slope of the line is \(-\frac{4}{3}\).
### Interpretation of the Slope:
The slope \(-\frac{4}{3}\) indicates a negative relationship between the x-values and y-values of the points. Specifically, for every increase of 3 units in the \(x\) direction, the \(y\) value decreases by 4 units. In practical terms, this means that as you move to the right along the x-axis, the line goes down at a rate of 4 units down for every 3 units you move right. This signifies a downward slant of the line from left to right.