Kids on a field trip to a dairy farm are gathered in the barn with cows. If there are 11 heads and 28 legs in the barn, how many kids and how many cows are
in the barn?

There were 8 kids and 3 cows in the barn.

These types of questions can be very confusing to grasp if you don’t know what to look for. In this case you need to really think about the words given.

When talking about heads, a cow and a kid have the same amount (one of course). So when the question states we have a total of 11 heads, we also have a total of 11 cows and kids combined. We can state this as x+y=11 (x being for the amount of kids and y being the amount of cows.) we can now put this equation to the side for now and find the next equation needed.

We then need to use the given legs in the barn, in which case kids have 2 and cows have 4, therefore we can get the equation 2x+4y=28 (x being the amount of kids and y being the amount of cows).

We now have two options to do complete this task: elimination and substitution. We’ll do substitution as to I confuse you too much.

So we have two equations x+y=11 and 2x+4y=28. We need to substitute in our value for x or y. So to do this I will find what x equals from the first equation we made (x+y=11). First subtract y from both sides of the equation to get x=11-y.

Next (so we can make this even simpler) let’s grab the equation 2x+4y=28 and make it simplified even more by halving the entire equation because all the numbers are even, leaving us with x+2y=14.

Now here’s the substitution part, since we have the first equation as to what x would equal (x=11-y) we then put that into the new equation x+2y=14. The equation that will solve for y while now look like 11-y+2y=14. Now we want to add like terms to get 11+y=14 and then subtract by 11 to get y by itself which leaves us with y=3. Y is stated in our equation to be the amount of cows, therefore we have 3 cows. Because we had a total of 11 cows and kids, we can then infer there are 8 kids in the barn based on the previous equation x+y=11 now being x+3=11.

The vertex of the quadratic function is (1.5, -.75)