Fertilizer is sold in 100 pound bags labelled with the amount of nitrogen (N), phosphoric acid ( P205), and potash (K20) present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a 100 pound bag of Vigoro Ultra Turf fertilizer contains 29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash. Another type of fertilizer, Parkerâdos Premium Starter, has 18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash per 100 pounds. Determine the amount of each type required to yield a mixture containing the 112 pounds of nitrogen, 81 pounds of phosphoric acid, and 26 pounds of potash. The mixture contains pounds of Vigoro, and !!! pounds of Parkerâ D s. Hint: Let x1 be the amount of 100 pound bags of Vigoro, and X, the amount of 100 pound bags of Parker. Set up and solve the system in the unknowns X1 and X2 and then multiply the solution by 100 to find the number of pounds.

We need to find a mixture containing 112 pounds of nitrogen. A 100 pound bag of Vigoro contains 29 pounds of nitrogen, and a 100 pound bag of Parker has 18 pounds of Nitrogen. So:

[tex]29x_{1} + 18x_{2} = 112[/tex]

We need to find a mixture containing 81 pounds of phosporic acid. A 100 pound bag of Vigoro contains 3 pounds of phosporic acid, and a 100 pound bag of Parker has 25 pounds of phosporic acid. So:

[tex]3x_{1} + 25x_{2} = 81[/tex]

We need to find a mixture containing 26 pounds of potash. A 100 pound bag of Vigoro contains 4 pounds of potash, and a 100 pound bag of Parker has 6 pounds of potash. So:

[tex]4x_{1} + 6x_{2} = 26[/tex]

Solving the system

We have the following system

[tex]1)29x_{1} + 18x_{2} = 112[/tex]

[tex]2)3x_{1} + 25x_{2} = 81[/tex]

[tex]3)4x_{1} + 6x_{2} = 26[/tex]

I am going to solve this system by substitution. I am going to write [tex]x_{2}[/tex] as a function of [tex]x_{1}[/tex] in equation 3) and replace it in equation 1).

So

[tex]4x_{1} + 6x_{2} = 26[/tex]

[tex]6x_{2} = 26 - 4x_{1}[/tex]

[tex]x_{2} = \frac{26 - 4x_{1}}{6}[/tex]

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[tex]29x_{1} + 18x_{2} = 112[/tex]

[tex]29x_{1} + 18\frac{26 - 4x_{1}}{6} = 112[/tex]

[tex]29x_{1} + 3(26 - 4x_{1}) = 112[/tex]

[tex]29x_{1} + 78 - 12x_{1} = 112[/tex]

[tex]17x_{1} = 34[/tex]

[tex]x_{1} = 2[/tex]

We are going to need 2 100 pound bags of Vigoro = 200 pounds of Vigoro

[tex]x_{2} = \frac{26 - 4x_{1}}{6} = \frac{26 - 8}{6} = 3[/tex]

We are going to need 3 100 pound bags of Parker = 300 pounds of Parker

We now replace the values in the equation 2) to verify that the system is consistent.

[tex]3x_{1} + 25x_{2} = 81[/tex]

[tex]3(2) + 25(3) = 81[/tex]

[tex]6 + 75 = 81[/tex]

[tex]81 = 81[/tex]

The system is consistent, so the answer that we found is correct.