Answer:
10 gallons of chemical x, 5 gallons of chemical y, and 12 gallons of chemical z
Step-by-step 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:
[tex]\left[\begin{array}{ccc}1/5&0&1/3\\2/5&0&1/3\\2/5&1&1/3\end{array}\right][/tex] × [tex]\left[\begin{array}{ccc}x\\y\\z\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}6\\8\\13\\\end{array}\right][/tex]
Multiply the inverse of the 3x3 matrix by the "result" matrix:
[tex]\left[\begin{array}{ccc}-5&5&0\\0&-1&1\\6&-3&0\end{array}\right][/tex]×[tex]\left[\begin{array}{ccc}6\\8\\13\\\end{array}\right][/tex]= [tex]\left[\begin{array}{ccc}10\\5\\12\\\end{array}\right][/tex]
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.