How do you work out 5x plus 7y equals 89?

Answer 1

Presumably, x and y have to be positive integers.

If 5x + 7y = 89 then we need to find a multiple of 7 that when subtracted from 84 leaves a number ending in 0 or 5, thus allowing this remainder number to be divisible by 5.

7 x 1 = 7 : 89 - 7 = 82

7 x 2 = 14 : 89 - 14 = 75........which meets the above requirement

Then, 5x = 75 : x = 15.

After this first multiple of 7 has been determined, further multiples of 7 must be formed from an increase of 5, 10, 15 etc to enable the remainder to be divisible by 5. In a similar way, further multiples of 5 must REDUCE by multiples of 7. This means that there are only two further values for x, namely x = 15 - 7 = 8 and x = 15 - 14 = 1.

So, 7 x (2 + 5) = 7 x 7 = 49 : 89 - 49 = 40 then 5x = 40, : x = 8

And, 7 x (2 + 10) = 7 x 12 = 84 : 89 - 84 = 5 then 5x = 5 : x = 1.

The three possible solutions are (x = 15, y = 2), (x = 8, y = 7) and (x = 1, y = 12).

As mentioned above, note that the values for x decrease by 7 and the values for y increase by 5 for successive solutions.