# Chapter 3 - Linear Systems - 3-2 Solving Systems Algebraically - Practice and Problem-Solving Exercises - Page 146: 21

$3$ 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.

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