Question

1. you are planning a training program to run a marathon. you start your program by running 5 miles per week and want to add 3.9 more miles to your program each week.

a. write an equation that represents the distance you run in miles, y, given x number of weeks.
b. graph the equation.
c. identify the slope and y - intercept of the graph. explain what each means in terms of the situation.
d. after training for 5 weeks, how many miles will you be able to run?

Explanation:

Step1: Determine the equation type

This is a linear relationship. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the y - intercept (initial value).

Step2: Identify the values of \(m\) and \(b\)

The initial value \(b = 5\) (miles run in week 0). The rate of change \(m=3.9\) (miles added per week). So the equation is \(y = 3.9x+5\).

Step3: Graph the equation

• Find two points:
• When \(x = 0\), \(y=3.9\times0 + 5=5\). So the point \((0,5)\) is on the line.
• When \(x = 10\), \(y=3.9\times10+5=39 + 5=44\). So the point \((10,44)\) is on the line.
• Plot the points \((0,5)\) and \((10,44)\) and draw a straight line through them.

Step4: Identify slope and y - intercept

The slope \(m = 3.9\). In the context of the problem, it means that the number of miles run per week increases by \(3.9\) miles each week. The y - intercept \(b = 5\). It means that at the start of the training program (week \(x = 0\)), the person runs \(5\) miles.

Step5: Calculate the distance after 5 weeks

Substitute \(x = 5\) into the equation \(y=3.9x + 5\).

$$y=3.9\times5+5=\frac{39}{10}\times5+5=\frac{195}{10}+5 = 19.5+5=24.5$$

Answer:

a. \(y = 3.9x + 5\)
b. Plot points \((0,5)\) and \((10,44)\) and draw a line.
c. Slope \(m = 3.9\) (miles added per week), y - intercept \(b = 5\) (initial miles).
d. \(24.5\) miles