The answer is spray x: 10, spray y: 5, spray z: 12

a – number of gallons of chemical A
b – number of gallons of chemical B
c – number of gallons of chemical C

x – number of gallons of spray x
y – number of gallons of spray y
z – number of gallons of spray z

CHEMICAL         SPRAY X          SPRAY Y          SPRAY Z
    a = 6          =        1/5x          +       0y          +          1/3z
    b = 8          =        2/5x         +       0y          +          1/3z
    c = 13         =        2/5x          +         y          +          1/3z

So:

1/5x + 1/3z = 6
2/5x + 1/3z = 8
2/5x + y + 1/3z = 13
____________
Multiply the first equation by (-1) and add it to the second equation:ž

– 1/5x – 1/3z = – 6
2/5x + 1/3z = 8
____________
1/5x = 2
x = 2 * 5 = 10

From here:

1/5x + 1/3z = 6
x = 10
___________
1/5 * 10 + 1/3z = 6
2 + 1/3z = 6
1/3z = 4
z = 4 * 3 = 12

1/5x + y + 1/3z = 13
x = 10
z = 12
______
2/5 * 10 + y + 1/3 * 12 = 13
4 + y + 4 = 13
8 + y = 13
y = 13 – 8 = 5