how many shots of each kind?

Question

a basketball team recently scored a total of 91 points on a combination of 2-points field goals, 3-point field goals, and 1-point foul shots. altogether the team made 51 baskets and 14 more 2-pointers than foul shots. how many shots of each kind were made?

Answer 1

Let's set up some variables.

T - total points

x - number of 1-point foul shots

y - number of 2-point field goals

z - number of 3-point field goals

Now let's set up some equations for how this basketball team scored a total of 91 shots

We know that they made 51 baskets total. We have three different types of baskets or shots. So that would mean that if you add up all the different kinds of shots, you will get 51. So...

x + y + z = 51

We also know that there were 14 more 2-pointers than foul shots, so let's set up that equation

y = 14 + x

Lastly, since there are 3 unknowns, that means we need three total equations. The next equation we can use is how many total points the team scored. We know how many points each basket scores for a team, and that they scored 91 total points, so let's set up the last equation.

1x + 2y + 3z = 91

Now that we have our three equations...

x + y + z = 51

y = 14 + x

x + 2y + 3z = 91

We can either solve the system algebraically or we can use matrices to solve for it. It depends on what you like doing better. Solving for the system gives

x = 16 (16 made foul shots)

y = 30 (30 made 2-point shots)

z = 5 (5 made 3-point shots)