Using the digits 1 to 9 at most one time each, fill in the boxes to find the largest (or smallest) possible values for x.

smallest: 8x -3 = 4; 1y +9 = 2. total = -49/8

 largest: 1x -9 = 8; 2y +3 = 7. total = 19

Step-by-step explanation:

If we use variables to represent the box contents, we can write ...

ax -b = c

dy +e = f

Then the values of x and y are ...

 x = (c +b)/a

 y = (f -e)/d

For positive integer values of the variables, x will always be positive, and y may or may not be negative.

Smallest sum

For the sum to be the smallest, we must have x be as small as possible and the ratio (f-e)/d be as negative as possible.

x will be small for large 'a' and for (c+b) small. For y to be as negative as possible, we want 'd' and 'f' small and 'e' large. Best results are obtained for

8x -3 = 4   ⇒   x = 7/8

1y +9 = 2   ⇒   y = -7

For these coefficients, the sum is -6 1/8 = -49/8.

(note that the values of 'b' and 'c' can be swapped with no net effect)

Largest sum

For the sum to be the largest, we must have x as large as possible: (b+c) large and 'a' small. At the same time we must have y be positive and as large as possible: (f-e) positive and large, 'd' small. Best results are obtained for

1x -9 = 8  ⇒   x = 17

2y +3 = 7   ⇒   y = 2

For these coefficients, the sum is 19. Again, 'b' and 'c' can be swapped with no effect.

Additional comment

These extreme values are verified by examination of the 60,480 possible permutations of the coefficients.

answer: 1 and 3

step-by-step explanation: