Why must two parallel lines have different y-intercepts?
Two parallel lines must have different y-intercepts because the y-intercept represents the point where the line crosses the y-axis. Since the lines are parallel, they will never intersect, meaning they will never cross the y-axis at the same point. Consequently, their y-intercepts will be different.
Two parallel lines must have different y-intercepts because the y-intercept represents the point where a line intersects the y-axis. In order for two lines to be parallel, they must have the same slope but different y-intercepts.
To understand why parallel lines require different y-intercepts, we can consider the equation of a line in slope-intercept form: y = mx + b. In this equation, "m" represents the slope of the line, and "b" represents the y-intercept.
If two lines have the same slope (m1 = m2), and the same y-intercept (b1 = b2), then the equation of both lines would be y = mx + b, resulting in the same line. This means they would intersect at every point, and they would not be parallel.
However, if the y-intercepts are different (b1 ≠ b2), even though the slopes are the same, the equations of the lines would be y = mx + b1 and y = mx + b2. These lines will never intersect because they are always a constant vertical distance apart. Hence, they remain parallel.
So, to summarize, if two lines have the same slope and different y-intercepts, they will be parallel because they will never intersect.