Final answer:
To find the slope and y-intercept of a linear equation, one must put the equation in slope-intercept form (y = mx + b). The slope (m) is the change in y over the change in x, and the y-intercept (b) is where the line crosses the y-axis. Comparing slopes determines if lines are parallel, perpendicular, concurrent, or coincident.
Step-by-step explanation:
The question involves finding the slope and y-intercept of a linear equation and determining the relationship between two graphs, such as whether they are parallel, perpendicular, concurrent, or coincident. To find the slope and y-intercept of a linear equation, you can write it in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. The slope indicates the steepness of the line and is calculated as the rise over the run (the change in y over the change in x). The y-intercept is the point where the line crosses the y-axis.
To determine the relationship between two graphs, compare their slopes:
- If the slopes are identical, and the y-intercepts are the same, the lines are coincident (the lines lie on top of each other).
- If the slopes are identical, but the y-intercepts are different, the lines are parallel.
- If the slopes are negative reciprocals of each other (the product of the slopes is -1), the lines are perpendicular.
- If the slopes are neither identical nor negative reciprocals, the lines are concurrent (intersecting but not perpendicular).
This analysis is essential in understanding how to interpret the equation of a line and how to manipulate a line, whether for algebraic purposes or in the context of graphical analysis of one-dimensional motion.