Final answer:
The point-slope form (y - y1 = m(x - x1)) is used when the slope and a single point on the line are known, while the slope-intercept form (y = mx + b) directly shows the slope and y-intercept, which is useful for quickly graphing lines.
Step-by-step explanation:
The difference between the formulas y - y1 = m(x - x1) and y = mx + b lies in their applications and representations of a line in coordinate geometry. The equation y - y1 = m(x - x1) is known as the point-slope form of a linear equation, and it defines a line based on the slope (m) and a specific point (x1, y1) that lies on the line.
This form is particularly useful when you need to find the equation of a line when you're given the slope and just one point or when you're working with two points and need to calculate the slope.On the other hand, the equation y = mx + b is known as the slope-intercept form of a linear equation. This form directly shows the y-intercept (b), which is where the line crosses the y-axis, and the slope (m).
The y-intercept is the value of y when x is zero, and the slope indicates the steepness of the line, defined as the rise over run. The slope-intercept form provides a quick way to graph the line simply by starting at the y-intercept and using the slope to find additional points.
In summary, while both equations describe a straight line, they are used in scenarios based on the information available and the particular needs of the situation. When plotting a line with slope three and y-intercept nine, as represented by the equation y = 9 + 3x, one can see how these constants define the line's properties and behavior on a graph.