Does this help you to conclude anything about the linear equation y = ax + b when a is fixed but b varies?
When 'a' is fixed and 'b' varies in the equation y = ax + b, the slope of the line remains constant, but the y-intercept changes. The line cuts the y-axis at the point (0, b), so varying 'b' shifts the point where the line intersects the y-axis.
As a result, changing the value of 'b' shifts the line while keeping it parallel to the original line. Lines with equal slopes but different y-intercepts are parallel to each other. For example, the equations y = 2x - 1, y = 2x + 1, and y = 2x + 5 have the same slope but different y-intercepts, making them parallel lines.
Explanation
The textbook explicitly concludes that in the equation y = ax + b, 'a' represents the slope and 'b' represents the y-intercept. When 'b' varies while 'a' remains fixed, the slope stays the same but the y-intercept shifts. This causes the lines to shift position while remaining parallel to the original line, as lines with equal slopes but different y-intercepts are always parallel.