The water park is a popular field trip destination. This year the senior class at high school a and high school b both planned trips there. The senior class at high school a rented filled 4 vans and 14 buses with 742 students. High school b rented and filled 4 vans and 6 buses with 350 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.

So let’s start of by setting variables for buses and vans
x = buses
y = vans

our equations will look something like this :

4x + 14y = 742 &’
4x + 6y = 350

let’s solve for y first . we need to cancel out our x’s so we turn on of them into a negative and since both of them have the same common factor we don’t need to find it . let’s multiply the bottom equation by ( -1 )

here’s what our new equations look like :
4x + 14y = 742
-4x - 6y = -350

now that our x’s are cancelled out we add our y’s and our totals :

14y + ( -6y ) = 8y
742 + ( -350 ) = 392

we want y alone so we divide :

392/8 = 49

y = 49

now we can substitute 49 for y :

4x + 14 ( 49 ) = 742
4x + 6 ( 49 ) = 350

we can multiply those numbers together :

4x + 686 = 742
4x + 294 = 350

now we add downward :

8x + 980 = 1,092

we need x alone so we subtract 980 from both sides :

8x = 112

divide by 8 :

x = 14

now , we can substitute 14 for x and 49 for y to check our work :

4 ( 14 ) + 14 ( 49 ) = 742

742 = 742 ✅

4 ( 14 ) + 6 ( 49 ) = 350

350 = 350 ✅