Answer
$3$ vans and $2$ sedans.
Work Step by Step
Let
$v = $ number of vans
$s= $ number of sedans.
Knowing there are five vehicles, then
$v + s = 5$ (Equation 1)
Knowing there are a total of $31$ people, with vans carrying seven and sedans five:
$7v + 5s = 31$ (Equation 2)
Thus, the system that models the situation is:
$v+s=5$ (Equation 1)
$\\7v+5s=31$ (Equation 2)
Solve for $v$ by subtracting $s$ to both sides of Equation 1:
$v = 5-s$
Substitute $5 - s$ to $v$ in Equation 2:
$7v+5s=31
\\7(5-s) + 5s = 31$
Distribute $7$:
$35 - 7s + 5s = 31$
Collect like terms:
$35 - 2s = 31$
Subtract $35$ to both sides of the equation:
$-2s =-4s$
Divide both sides by $-2$:
$s = 2$
Substitute $s = 2$ into Equation 1:
$v+s=5
\\v+2=5
\\v=5-2
\\v=3$
Thus, $v = 3$, $s = 2$
Therefore there should be $3$ vans and $2$ sedans.
