To solve this problem:

Given:

1. Total points = 22
2. Nathan made 4 times as many two-point shots as three-point shots.

Variables:

• Let $x$ = number of three-point shots.
• Let $y$ = number of two-point shots.

Equations:

1. Total points equation: $3x + 2y = 22$
2. Relationship between shots: $y = 4x$

Substitution:

Substitute $y = 4x$ into the total points equation: $3x + 2(4x) = 22$ $3x + 8x = 22$ $11x = 22$ $x = 2$

Using $y = 4x$: $y = 4(2) = 8$

Solution:

• Nathan made 2 three-point shots.
• Nathan made 8 two-point shots.

Would you like further details or clarification? Here are some related questions to expand on this concept:

1. How would the solution change if Nathan scored 30 points instead of 22?
2. Can you derive a formula for cases where the ratio between two-point and three-point shots changes?
3. What if the number of two-point shots was 3 times the number of three-point shots?
4. How would you set up this problem using matrices?
5. Can you generalize this solution for any total score $S$ and ratio $r$?

Tip: Always check your solution by substituting the values back into the original equations!