Solution:
Given:
[tex]\begin{gathered} \text{Total students that went for the field trip= 231 students} \\ \text{Vans can hold 7 stduents each} \\ \text{Buses can hold 25 students each} \\ \text{Total buses and vans =15} \end{gathered}[/tex]Question 7a:
Let
[tex]\begin{gathered} v\text{ represent vans} \\ b\text{ represent buses} \end{gathered}[/tex]The following equations can be deduced from the question;
[tex]\begin{gathered} \text{Total students that the total number of vans can hold is;} \\ 7\times v=7v \\ \\ \text{Total students that the total number of buses can hold is;} \\ 25\times b=25b \\ \\ \text{Total students on the trip is 231, hence,} \\ 7v+25b=231\ldots\ldots\ldots\ldots\ldots.(1) \\ \\ \\ \text{Also,} \\ \text{Total vans and buses is 15. This means;} \\ v+b=15\ldots\ldots\ldots\ldots\ldots\ldots\text{.}(2) \end{gathered}[/tex]Therefore, the system of equations to represent this situation is;
[tex]\begin{gathered} 7v+25b=231 \\ v+b=15 \end{gathered}[/tex]Ha
Question 7b.
Solving equations (1) and (2) simultaneously,
[tex]\begin{gathered} 7v+25b=231\ldots\ldots\ldots\text{.}\mathrm{}(1) \\ v+b=15\ldots\ldots\ldots\ldots\ldots\text{.}(2)\times7 \\ 7v+7b=105\ldots\ldots\ldots\ldots(3) \\ \\ \text{Subtracting equation (3) from (1),} \\ equation(1)-equation(3); \\ 7v-7v+25b-7b=231-105 \\ 18b=126 \\ \text{Dividing both sides by 18 to get b,} \\ b=\frac{126}{18} \\ b=7 \\ \\ \\ To\text{ get v, substitute b in euqation (2)} \\ v+b=15 \\ v+7=15 \\ v=15-7 \\ v=8 \\ \\ \\ \text{Hence, } \\ 7\text{buses and 8 vans went on the field trip} \end{gathered}[/tex]Therefore, DORS needs 7 buses for the field trip.