Question

The object of the 24 game, created by Robert Sun, is to combine four numbers, using addition, subtraction, multiplication and or division to get the number 24. For examples the numbers 2,5,5,4 can be combined as 2(5+5)+4= 24. For the algebra edition of the game and the game card shown to the right, the object is to find single digit positive integer value x and y so the four numbers x+y, 3x+2y, 8 and 9 can be combined to make 24. Using the game card, write a system of equations that, when solved, can be used to make 24 from the game card. what is the solution to this system, and how can it be used to make 24 on the game card?

Answer 1

4x +3y = 7; 1 ≤ x ≤ 9; 1 ≤ y ≤ 9

(x, y) = (1, 1)

Explanation:

If you simply add the numbers, you have ...

(x+y) +(3x+2y) +8 +9 = 24

This simplifies to ...

4x +3y = 7

Perhaps the other "equations" in the "system of equations" are the constraint inequalities ...

• 1 ≤ x ≤ 9
• 1 ≤ y ≤ 9

The smallest possible value of both x and y is 1, which would satisfy this equation:

4·1 + 3·1 = 7

Using the expression (x+y) +(3x+2y) +8 +9 = 24, suitable values for x and y are 1 and 1.

_____

We have written one equation, which in this context only has one solution for the two variables. It seems likely that we need to know more about the game card in order to write a system of equations.