Eva maintained an average speed of $v_1$ miles per hour for the first $t_1$ hours of her road trip. For the next $t_2$ hours of the trip, she drove at an average speed of $v_2$ . Eva drove a total of $D$ miles in $T$ hours. Which of the following systems of equations could be used to find how many miles Eva drove in the first $t_1$ hours of the trip?

Full solution

Q. Eva maintained an average speed of

v_1

miles per hour for the first

t_1

hours of her road trip. For the next

t_2

hours of the trip, she drove at an average speed of

v_2

. Eva drove a total of

D

miles in

T

hours. Which of the following systems of equations could be used to find how many miles Eva drove in the first

t_1

hours of the trip?

Denote Average Speeds and Times: Let's denote the average speed for the first part of the trip as $v_1$ (in miles per hour), the time for the first part as $t_1$ (in hours), the average speed for the second part as $v_2$ , and the time for the second part as $t_2$ . The total distance is given as $D$ and the total time as $T$ . We can set up two equations: one for the distance covered in each part of the trip, and one for the total time of the trip.
Equation for Distance Covered: The distance Eva drove in the first part of the trip can be represented as $v_1 \times t_1$ , and the distance in the second part as $v_2 \times t_2$ . The total distance is the sum of these two distances, so we have the equation $v_1 \times t_1 + v_2 \times t_2 = D$ .
Equation for Total Time: The total time of the trip is the sum of the time spent on each part, so we have the equation $t_1 + t_2 = T$ .
Solving the System of Equations: Now we have a system of two equations with two variables ( $t_1$ and $t_2$ ) that can be solved to find the distance Eva drove in the first part of the trip. The system of equations is: $\newline$ $1$ ) $v_1 \times t_1 + v_2 \times t_2 = D$ $\newline$ $2$ ) $t_1 + t_2 = T$