Answer:
There are 24 students and teachers that rode in vans, and there are 385 students and teachers that rode in buses.
Step-by-step explanation:
First we are going to set up a system of equations where:
<em>b= number of buses</em>
<em>v= number of vans</em>
We know from the equation that each bus transported 55 people and that each van transported 12 people and that there was a total of 409 people transported. We also know that the number of buses was 5 more than the number of vans.
Knowing that we can set up our system:

Since we know what <em>b</em> is equal to, we can now use the substitution method to plug the value of <em>b</em> from the bottom equation into the one on the top and solve for variable <em>v.</em>
55(<em>v </em>+5)+12<em>v </em>=409
From there we can distribute 55 into (v+5)
55<em>v </em>+275+12<em>v </em>=409
We can now combine like terms.
67<em>v</em> +275 = 409
From there we subtract 275 from both sides of the equation to isolate the variable.
67<em>v +</em>275 = 409
-275 -275
After doing this, we now divide both sides of the equation by 67 to solve for <em>v.</em>

We now know that <em>v, </em>the number of vans is 2 and can plug that into our bottom equation from our system of equations (
).

Knowing that there are 7 buses and 2 vans, we can now plug these numbers into the terms in the top equation of our system. <em>55b </em><em>and </em><em>12v.</em>


385 people rode buses. 24 people rode vans
Check your equation
409 people went in total. 385 + 24 = 409.