Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'.
y = 4x, y = 2x, y = x
For y = 4x, y = 2x, y = x, the value of b is 0, so all lines pass through the origin (0, 0). As the value of a changes (4, 2, 1), the slope changes. Since a > 1, the lines are steeper than y = x.
y = -6x, y = -3x, y = -x
For y = -6x, y = -3x, y = -x, the value of b is 0, so all lines pass through the origin (0, 0). As the value of a changes (-6, -3, -1), the slope changes. The lines slope downwards due to negative a.
y = 5x, y = -5x
For y = 5x, y = -5x, the value of b is 0, so both lines pass through the origin (0, 0). The slopes are a = 5 and a = -5. They are equally steep but go in opposite directions.
y = 3x - 1, y = 3x, y = 3x + 1
For y = 3x - 1, y = 3x, y = 3x + 1, the value of a is fixed at 3, so the slope remains the same. As b changes (-1, 0, 1), the y-intercept changes, causing the lines to shift but remain parallel to each other.
y = -2x - 3, y = -2x, y = 2x + 3
For y = -2x - 3, y = -2x, y = 2x + 3, the first two lines have the same slope (a = -2) but different y-intercepts (b = -3, 0), making them parallel. The third line has a different slope (a = 2) and is not parallel.
Explanation
In the equation y = ax + b, 'a' represents the slope of the line and 'b' represents the y-intercept. If we change the values of 'a', keeping 'b' fixed, the slope of the line changes while the y-intercept remains fixed. If we change the values of 'b' keeping the value of 'a' fixed, the lines shift but remain parallel to the original line. Thus, lines with equal slopes but different y-intercepts are parallel to each other.