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:
b= number of buses
v= number of vans
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:
[tex]\left \{ {{55b+12v=409} \atop {b=v+5\\}} \right.[/tex]
Since we know what b is equal to, we can now use the substitution method to plug the value of b from the bottom equation into the one on the top and solve for variable v.
55(v +5)+12v =409
From there we can distribute 55 into (v+5)
55v +275+12v =409
We can now combine like terms.
67v +275 = 409
From there we subtract 275 from both sides of the equation to isolate the variable.
67v +275 = 409
-275 -275
After doing this, we now divide both sides of the equation by 67 to solve for v.
[tex]\frac{67v}{67} =\frac{134}{67}[/tex]
We now know that v, the number of vans is 2 and can plug that into our bottom equation from our system of equations ([tex]b=v+5[/tex]).
[tex]b=2+5\\b=7[/tex]
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. 55b and 12v.
[tex]55(7)=385[/tex]
[tex]12(2)= 24\\[/tex]
385 people rode buses. 24 people rode vans
Check your equation
409 people went in total. 385 + 24 = 409.
