Final Answer:
The equation for the line with a slope (m) of 3 and a y-intercept of (0, 2) is A) y = 3x + 2.
Thus option a is correct.
Step-by-step explanation:
To determine the equation of a line given its slope and y-intercept, we use the slope-intercept form of a line, y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. Here, the slope is given as m = 3, and the y-intercept is (0, 2). The y-intercept is the point where the line crosses the y-axis, which gives us the value of 'b'. So, the equation becomes y = 3x + 2.
Explanation (180-250 words):
To find the equation of a line with a given slope and y-intercept, we utilize the slope-intercept form of a line, y = mx + b. In this scenario, the slope 'm' is provided as 3, indicating how steep the line is. Additionally, the y-intercept represents the point (0, 2) where the line intersects the y-axis. This intercept provides us with the value of 'b', the constant term in the equation. Thus, by substituting the slope (m = 3) and the y-intercept's y-coordinate (b = 2) into the equation form, we derive the line's equation as y = 3x + 2.
The slope-intercept form, y = mx + b, delineates the relationship between 'x' and 'y' on a Cartesian plane. The slope 'm' denotes the rate of change of 'y' concerning 'x', implying that for every unit change in 'x', the 'y' value changes by the slope value. Meanwhile, the y-intercept 'b' determines the point where the line intersects the y-axis. In this case, the slope of 3 implies that for every unit change in 'x', 'y' increases by 3 units, while the y-intercept of 2 positions the line 2 units above the origin along the y-axis. Hence, combining these aspects yields the equation y = 3x + 2, representing the line's behavior and position on the Cartesian plane.
Therefore option a is correct.