Question

In basketball there are three ways to score. 1 point for a free Throw, 2 points for a short field goal, and 3 points for a long field goal. During the first quarter of a regular season basketball game, the Dallas Mavericks scored 36 points but did not make any free throws. What is the other extreme case in table 1.1 the entry where SFG=0 and how can you systemically record the other entries in table 1.1

Answer 1

It says Dallas didn't make any free throws. It means there are short field goals (SFG) and long field goals (LFG) only.
Let's assume number of SFG = x
and number of LFG = y

Given that 2 points for a short field goal, and 3 points for a long field goal.
Total Score = 2x+3y
Also, Total score = 36 (given in the question)
So we have the equation
$2x + 3y = 36$
where x and y are integers.

when SFG=0, then x=0 and so 3y=36.
y=12 i.e. LFG = 12
So the extreme case is (0,12) i.e. SFG=0 and LFG=12.

Other cases would be integral vales of (x,y) which satisfy the equation 2x+3y=36.
So (3,10) , (6,8) , (9,6) , (12,4) , (15,2) , and (18,0) are other cases.

Hence final answers are (x,y) = (0,12) , (3,10) , (6,8) , (9,6) , (12,4) , (15,2) , and (18,0)