To design a fair race, we need to create a system of linear equations that represents the distance each animal can travel. In this case, we have a coyote that runs at 43 mph and a rabbit that runs at 35 mph. Let's define some variables:
Let x represent the distance (in miles) traveled by the coyote.
Let y represent the distance (in miles) traveled by the rabbit.
Since we want both animals to have an equal chance of winning if they race at their top speed, the race is fair when they both finish the race in the same amount of time.
To determine the time it takes for each animal to travel their respective distances, we can use the formula:
Time = Distance / Speed
For the coyote:
Time = x / 43
For the rabbit:
Time = y / 35
Since the race is fair, we want both animals to finish in the same amount of time. Therefore, we can set up the following equation:
x / 43 = y / 35
This is our first equation in the system.
To create the second equation, we can consider the fact that both animals start at the same time and finish the race at the same time. This means that the total time for both animals to complete the race is the same.
We can calculate the time it takes for the coyote as:
Time = Distance / Speed
Time = x / 43
Similarly, for the rabbit, the time can be calculated as:
Time = Distance / Speed
Time = y / 35
To represent the total time for both animals finishing the race, we add the times together:
x / 43 + y / 35
Since we know the total time should be the same for both animals, we can set up the second equation:
x / 43 + y / 35 = t
Here, t represents the total time for both animals to complete the race. This is our second equation in the system.
Therefore, the system of two linear equations representing the distance each animal can travel to model the fair race is:
Equation 1: x / 43 = y / 35
Equation 2: x / 43 + y / 35 = t
Be sure to substitute the appropriate variables and values into the equations based on the chosen speeds of the coyote and the rabbit.
Once you have the equations, you can solve them using algebraic methods such as substitution, elimination, or graphing to find the values of x and y that satisfy both equations.
Then you can proceed to graph the system to prove that the race is fair.