Final answer:
The slope of a line parallel to the equation y - x is 1. Slope indicates how y increases as x increases, and parallel lines have identical slopes. Thus, any line parallel to y - x rises 1 unit for every 1 unit increase along the x-axis, maintaining a constant rate of change.
Step-by-step explanation:
The slope of a line parallel to the line defined by the equation y − x is the same as the slope of the original line. In the case of the equation y − x, we can rearrange it to y = x + 0 to see that the slope (m) is 1, because it rises 1 unit for every increase of 1 unit along the x-axis. Therefore, a line parallel to y − x will also have a slope of 1.
A line's slope is important because it tells us how the dependent variable (y) changes in relation to changes in the independent variable (x). A positive slope like this suggests that as x increases, y also increases at a consistent rate. Since parallel lines have the same slope, any line parallel to y − x will mirror this behavior, maintaining a constant rise over run.
An example to further illustrate: if we take another line y = x + 5, this line has the same slope of 1, but a different y-intercept. It's parallel to y − x because they both increase by 1 unit in the y direction for each 1 unit of increase in the x direction. This relationship is visualized graphically by showing that for any given x, both lines rise at the same rate, maintaining the same angle with respect to the x-axis.