I used an algebraic method to find the multiples using the following method:
For example,
a(x+y)+b(x+2y) = 5x+8y
{a and b represent the values of the multiples we need to find}
We can then form two equations if we substitute x=0 and y=0,
When x=0,
ay+2by = 8y
When y=0,
ax+bx = 5x
Then, we can cancel the like terms in each equation:
Equation 1: a+2b = 8
Equation 2: a+b = 5
Which we can solve simultaneously:
Equation 1 - Equation 2 (as 'a' is present in both)
b = 3
Substitute into Equation 2
a = 2
Therefore, 2 (x+y) +3 (x+2y) = 5x+8y
I repeated this for all other possibilities:
2 (x+y) +3 (x+2y) {from above}
6 (x+y) -1 (x-2y)
4 (x+y) +1 (x+4y)
1 (x+y) +2 (2x+3y)
6.5 (x+2y) -0.75 (x-2y)
2 (x+2y) +1.5 (x+4y)
1 (x+2y) +2 (2x+3y)
6 (x-2y) +0.5 (x+4y)
31/7 (x-2y) -2/7 (2x+3y)
17/5 (x+4y) +2/5 (2x+3y)