Question

Nigel and Phyllis train for a marathon. The equation y = 1.25z + 4 represents that Nigel currently runs 4 miles per week and increases his total miles per week, y, by 1.25 for a numberof weeks. The graph represents Phyllis's running mileage.

Answer 1

The student's question deals with representing linear equations graphically, the concept of slope and y-intercept, and the application of these concepts to real-world scenarios such as training schedules, stock prices, and labor charges.

Step-by-step explanation:

The student's question is related to linear equations and how to represent them graphically, which is a crucial part of algebra in high school mathematics. The provided information pertains to equations of the form y = mx + b, where m is the slope and b is the y-intercept. For instance, Nigel's running schedule is represented by the equation y = 1.25z + 4, signifying a starting point of 4 miles and an increase of 1.25 miles each week.

Best-fit lines and regression analysis are also covered in the question, with references to the least-squares regression line for certain datasets being given by an equation like ŷ = -173.51 + 4.83x, which is used to predict values and understand the relationship between variables.

The concept of slope and y-intercept is vital in understanding linear relationships. For example, in a business context, the equation y = 55x + 75 shows that the total labor charge (y) for fixing a car is proportional to the number of hours worked (x), with an initial fee of 75.

Answer 2

From Phyllis's graph:

`\begin{gathered} (x1,y1)=(0,6) \\ (x2,y2)=(4,9) \\ m=(y2-y1)/(x2-x1)=(9-6)/(4-0)=(3)/(4) \\ \text{ Using the point slope equation:} \\ y-y1=m(x-x1) \\ y-6=(3)/(4)(x-0) \\ y=(3)/(4)x+6 \\ y=0.75x+6 \end{gathered}`

At the start of training, Phyllis runs more miles per week than Nigel because Phyllis runs 6 miles per week, while Nigel runs 4 miles per week