In the context of slope of a line, the term 'run' refers to the horizontal distance or change in the x-coordinate between two points on a line. It is one of the key components, along with 'rise,' that is used to calculate the slope of a line.
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The run of a line is the change in the x-coordinate between two points, and it is represented by the symbol Δx (delta x).
The run of a line can be positive or negative, depending on the direction of the line (left to right or right to left).
The run is a key component in the slope formula, $\text{slope} = \frac{\text{rise}}{\text{run}}$, which is used to determine the steepness of a line.
The run of a line can be used to find the x-coordinate of a point on the line given the y-coordinate and the slope.
Understanding the concept of run is essential for interpreting and analyzing the slope of lines in the coordinate plane.
Review Questions
Explain how the run of a line is related to the slope of the line.
The run of a line is a crucial component in the calculation of the slope of the line. The slope formula is $\text{slope} = \frac{\text{rise}}{\text{run}}$, where the rise is the change in the y-coordinate and the run is the change in the x-coordinate between two points on the line. The run, along with the rise, determines the steepness or incline of the line, with a larger run corresponding to a smaller slope and a smaller run corresponding to a larger slope.
Describe how the sign of the run (positive or negative) affects the interpretation of the line's direction.
The sign of the run, whether positive or negative, indicates the direction of the line in the coordinate plane. If the run is positive, the line is sloping upward from left to right. If the run is negative, the line is sloping downward from left to right. The sign of the run, combined with the sign of the rise, determines the overall direction and quadrant of the line in the coordinate plane.
Analyze how the run of a line can be used to find the x-coordinate of a point on the line given the y-coordinate and the slope.
Since the slope formula is $\text{slope} = \frac{\text{rise}}{\text{run}}$, we can rearrange the formula to solve for the run: $\text{run} = \frac{\text{rise}}{\text{slope}}$. If we know the y-coordinate of a point on the line (the rise) and the slope of the line, we can use this relationship to determine the x-coordinate (the run) of that point. This allows us to locate specific points on the line based on the given information about the slope and the y-coordinate.