Question

a mixture 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?

Answer 1

10 gallons of chemical x, 5 gallons of chemical y, and 12 gallons of chemical z

Explanation:

Set up the system of equations:

1/5 x + 0 y + 1/3 z = 6

2/5 x + 0 y + 1/3 z = 8

2/5 x + 1 y + 1/3 z = 13

Arrange them as matrices:

`\left[\begin{array}{ccc}1/5&0&1/3\\2/5&0&1/3\\2/5&1&1/3\end{array}\right]` ×
`\left[\begin{array}{ccc}x\\y\\z\end{array}\right]` =
`\left[\begin{array}{ccc}6\\8\\13\\\end{array}\right]`

Multiply the inverse of the 3x3 matrix by the "result" matrix:

`\left[\begin{array}{ccc}-5&5&0\\0&-1&1\\6&-3&0\end{array}\right]`×
`\left[\begin{array}{ccc}6\\8\\13\\\end{array}\right]`=
`\left[\begin{array}{ccc}10\\5\\12\\\end{array}\right]`

This tells you that you need 10 gallons of chemical x, 5 gallons of chemical y, and 12 gallons of chemical z to make the mixture.

Answer 2

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