19 people on 8 benches with 1, 2 or 3 people. How many with only 2?

Question

I found a math problem for 2nd graders in Norway. The article only presented drawing a table of everything and simply looking at it, instead of a mathematical approach, like a formula. I'm not sure what kind of solution is needed, so I cannot google the type of math needed. It goes like this:

-There are eight benches in a park.
-On each of the benches, there are either 1, 2 or 3 people. So no bench is empty.
-All together there are 19 people on the benches

How many benches have only two people?

The answer is 5: http://tinyimg.io/i/1SJcOzb.png

Is there a way to figure this out with a formula?

Answer 1

If there are $x$ benches with one person, $y$ with two and $z$ with three then you have the simultaneous equations $$x+y+z=8$$ $$x+2y+3z=19$$

You can eliminate one term to give (depending on which is eliminated)$$y +2x = 5$$ $$y+2z=11$$ $$z-x=3$$

so if $x,y,z$ are non negative integers, then from the first of these $x$ can only be any of $0,1,2$, requiring $z=3,4,5$ and $y=5,3,1$ respectively

Answer 2

But it's not true as the problem is stated. The stated solution is listed as

\begin{array}{r|cccccccc|} \text{Benches} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ \text{people} & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ & \bigcirc & \bigcirc & \bigcirc \\ \hline \end{array}

And there are five benches with only two people.

But what is wrong with this answer?

\begin{array}{r|cccccccc|} \text{Benches} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ \text{people} & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc\\ & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ \hline \end{array}

Or this answer?

\begin{array}{r|cccccccc|} \text{Benches} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ \text{people} & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc \\ & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \bigcirc\\ \hline \end{array}