Question

A

Answer 1

Answer:

1-point: 4

2-point: 9

3-point: 3

Step-by-step explanation:

Let x, y, z represent the numbers of 1-, 2-, and 3-point baskets, respectively. The given relations are ...

 x +2y +3z = 31 . . . . . . total points scored

 y -x = 5 . . . . . . . . . . . 5 more 2-point baskets than free throws

 y = 3z . . . . . . . . . . . . 3 times as many 2-point baskets as 3-point baskets

Using the second equation to write an expression for x, we have ...

 x = y -5

Substituting this and the third equation into the first, we get ...

 (y -5) +2y +(y) = 31

 4y = 36 . . . . . . . . . . . add 5, collect terms

 y = 9

 x = 9 -5 = 4

 z = y/3 = 9/3 = 3

The player made 4 free throws, 9 2-point baskets, and 3 3-point baskets.

_____

Alternate solution

The equations can be written in augmented matrix form as ...

 

Row-reducing this matrix using any of a number of calculators gives (x, y, z) = (4, 9, 3), as above.

Answer 2

Answer:

• 1-point: 4
• 2-point: 9
• 3-point: 3

Step-by-step explanation:

Let x, y, z represent the numbers of 1-, 2-, and 3-point baskets, respectively. The given relations are ...

  x +2y +3z = 31 . . . . . . total points scored

  y -x = 5 . . . . . . . . . . . 5 more 2-point baskets than free throws

  y = 3z . . . . . . . . . . . . 3 times as many 2-point baskets as 3-point baskets

Using the second equation to write an expression for x, we have ...

  x = y -5

Substituting this and the third equation into the first, we get ...

  (y -5) +2y +(y) = 31

  4y = 36 . . . . . . . . . . . add 5, collect terms

  y = 9

  x = 9 -5 = 4

  z = y/3 = 9/3 = 3

The player made 4 free throws, 9 2-point baskets, and 3 3-point baskets.

_____

Alternate solution

The equations can be written in augmented matrix form as ...

  $\left[\begin{array}{ccc|c}1&2&3&31\\-1&1&0&5\\0&1&-3&0\end{array}\right]$

Row-reducing this matrix using any of a number of calculators gives (x, y, z) = (4, 9, 3), as above.