Use distance = rate x time
For the jogging portion of the trip, 15 = xt, where x = the rate at which the person jogs and t is the amount of time the person spends jogging.
For the walking portion of the trip, 15 = (x + 2.5)(9 - t), where (x + 2.5) is the rate at which the person walks and (9 - t) is the amount of time it takes for the person to walk 15 miles.
We can use the Substitution Method to solve this system of equations. Solve the first equation for x.
x = 15/t
Now, substitute this value of x into the second equation.
15 (15/t + 2.5)(9 - t)
15 = 135/t - 15 + 45/2 - 5t/2
15t = 135 - 15t + 45t/2 - 5t²/2 (when we multiply both sides of the equation by t)
30t = 270 - 30t + 45t - 5t² (when we multiply both sides of the equation by 2)
270 - 15t - 5t² = 0
54 - 3t - t² = 0 (when we divide both sides of the equation by 5)
(9 + t)(6 - t) = 0
Since t represents time, it cannot be negative. Therefore, the only logical solution is t = 6 hours. To find the individual rates for each portion of the trip, solve the equation x = 15/t
x = 15/6 = 2.5 mph, which is the person's walking rate. The jogging rate would be x + 2.5 = 2.5 + 2.5 = 5 mph.
