Final answer:
Melanie and Tracy's equations will have the same slope but different y-intercepts, because while Melanie's x-values represent actual years, Tracy has redefined the starting point, making her x-values represent the number of years since 2010. This affects the position of the y-intercept without changing the rate at which the y-values increase with respect to x.
Step-by-step explanation:
Melanie and Tracy are using two different sets of ordered pairs to find their equations for a linear relationship. Melanie's ordered pairs are (2010, 48) and (2013, 59), while Tracy's are (0, 48) and (3, 59), with x representing the number of years since 2010 in Tracy's pairs. To compare the equations, let's calculate the slope (m) for each set of ordered pairs.
For Melanie's points:
Slope (m) = ∆y / ∆x = (59 - 48) / (2013 - 2010) = 11 / 3
For Tracy's points:
Slope (m) = ∆y / ∆x = (59 - 48) / (3 - 0) = 11 / 3
As we can see, both Melanie and Tracy have the same slope, indicating the rate of change between years and the associated value is consistent between both sets of points. However, the y-intercept will differ because Tracy has shifted the x-axis to start at 2010, making her y-intercept the same as the y-value of her first point (48). Melanie will have a different y-intercept because her x-values represent actual years. Thus, the two girls' equations will have the same slope but different y-intercepts.
Understanding the relationship between the slope and y-intercept is crucial in determining the shape of the line. In general, linear equations of the form 'y = mx + b' will have the slope 'm' determining the steepness or incline of the line and the y-intercept 'b' representing where the line crosses the y-axis.