Final answer:
The slope (m) of a linear equation y = mx + b is defined as the rise over run, reflecting the line's steepness. The y-intercept (b) is where the line crosses the y-axis. For a line with slope 3 and y-intercept 9, the slope means a unit increase on the x-axis results in a rise of 3 on the y-axis.
Step-by-step explanation:
The slope of a line is a measure of its steepness and is calculated as the rise divided by the run. When we have an equation of the form y = mx + b, m represents the slope, which is the coefficient of the x-term. The term b represents the y-intercept, which is the point where the line crosses the y-axis. If we have a line that has a slope of 3, it means that for every one unit increase in x, the value of y increases by 3. Similarly, if the y-intercept is 9, the line crosses the y-axis at the point (0, 9).
In figuring out the slope and y-intercept, if we're given two points (x1, y1) and (x2, y2), the slope m can also be calculated as (y2-y1)/(x2-x1). The y-intercept can often be visually identified in a graph where the line crosses the y-axis or calculated from the equation by setting x to 0 and solving for y.