Thanks, Cal !!!
Here's another way to solve this with linear programming
Set the equations up as equalities
2x + 5y =10 ⇒ 6x + 15y = 30
3x + 4y = 12 ⇒ -6x - 8y = -24 add these
7y = 6
y = 6/7
And
2x + 5(6/7) = 10
2x + 30/7 = 70/7
2x = 40/7
x = 20/7
These are the (x, y) values that will maximize 8x + 13y
And this max is 8(20/7) + 13(6/7) = [160 + 78 ] / 7 = 238/7 = 34
This can be confirmed with this graph : https://www.desmos.com/calculator/nwl5zh1adg
The max will occur at the "corner" point of the intersection of the above inequalities ⇒ (20/7, 6/7)