Answer:
Explanation:
To determine if the graphs of two linear equations are parallel, we need to compare their slopes. If the slopes are equal, then the lines are parallel. If the slopes are not equal, then the lines are not parallel.
In case (a), the two equations are y = x + 3 and y = x + 6. Both equations have a slope of 1, which means the lines have the same steepness. However, the y-intercepts are different (3 and 6), which means the lines are shifted up or down relative to each other. Since the slopes are equal, but the y-intercepts are different, the lines are parallel.
In case (b), the two equations are y = 4x - 1 and y = 1 - 4x. Both equations have a slope of -4, which means the lines have the same steepness. However, the y-intercepts are different (-1 and 1), which means the lines are shifted up or down relative to each other. Since the slopes are equal, but the y-intercepts are different, the lines are not parallel.
In case (c), the two equations are y = 2x - 3 and y = -2x + 3. The slopes of the two equations are 2 and -2, which are negative reciprocals of each other. This means the lines are perpendicular, not parallel.
In case (d), the two equations are 3y = x - 12 and 6y = 2x + 12. We can rewrite these equations in slope-intercept form by solving for y. The first equation becomes y = (1/3)x - 4 and the second equation becomes y = (1/2)x + 2. The slopes of the two equations are 1/3 and 1/2, which are not equal. Therefore, the lines are not parallel.