Ben participated in a race. during the first half of the race, he walked 3 miles and ran at a rate of 5 miles per hour for x hours. during the second half of the race, he walked for 2 miles and ran at a rate of 5.5 miles per hour for x hours. what equation could you write to solve for the number of hours ben ran during each half of the race? let y represent the number of miles in half the race. write two equations, one for the first half of the race and one for the second half. how could you transform these two equations to be the same as the original equation you wrote to solve for x? what do you think this means about solving for the variables in a situation represented by two equations?

The two equations are:

y = 5*x + 3

y = 5.5*x + 2

Step-by-step explanation:

By definition:

speed = distance/time

it can be rewritten as:

time = distance/speed

During the first half of the race Ben ran at a rate of 5 miles per hour for x hours and walked 3 miles. Let's call y, the number of miles in half the race; then he ran y-3 miles. The time he spend was:

x = (y-3)/5

Analogously, for the second half:

x = (y-2)/5.5

These two equations can be rewritten as

y = 5*x + 3

y = 5.5*x + 2

Which means that:

5*x + 3 = 5.5*x + 2

3-2 = 5.5*x -5*x

1 = 0.5*x

x = 2 hours

When you are solving for the variables in a situation represented by two equations you can isolate the same variable in both equations and equate the two terms, getting a new equation with only one variable. Another option is to isolate one variable in one equation and replace it in the other one.