Amixture of 6 gallons of chemical a, 8 gallons of chemical b, and 13 gallons of chemical c is required to kill a destructive crop insect. commercial spray x contains 1, 2, and 2 parts respectively, of these chemicals. commercial spray y contains only chemical c. commercial spray z contains chemicals a, b, and c in equal amounts. how much of each type of commercial spray is needed to get the desired mixture?

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

Solve for x over the real numbers:
2 x^2 - 4 x - 3 = 0

Divide both sides by 2:
x^2 - 2 x - 3/2 = 0

Add 3/2 to both sides:
x^2 - 2 x = 3/2

Add 1 to both sides:
x^2 - 2 x + 1 = 5/2

Write the left hand side as a square:
(x - 1)^2 = 5/2

Take the square root of both sides:
x - 1 = sqrt(5/2) or x - 1 = -sqrt(5/2)

Add 1 to both sides:
x = 1 + sqrt(5/2) or x - 1 = -sqrt(5/2)

Add 1 to both sides:
x = 1 + sqrt(5/2) or x = 1 - sqrt(5/2)