Fertilizer is sold in bags labeled with the amounts of nitrogen ([tex]N[/tex]), phosphoric acid ([tex]P_2O_5[/tex]), and potash ([tex]K_2O[/tex]) present. The mixture of these nutrients varies from one type of fertilizer to the next.

For example:
- A 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, Parker's Premium Starter, has 18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash per bag.

Determine the number of bags of each type required to yield a mixture containing 101 pounds of nitrogen, 103 pounds of phosphoric acid, and 28 pounds of potash.

35 bags of Vigoro Ultra Turf fertilizer and 20 bags of Parker's Premium Starter fertilizer are required to yield a mixture containing 101101 pounds of nitrogen, 103103 pounds of phosphoric acid, and 2828 pounds of potash.

To determine the number of bags of each type of fertilizer required to yield a specific mixture of nutrients, we can set up a system of equations based on the given nutrient content of each bag.

By solving these equations, we find that 35 bags of Vigoro Ultra Turf fertilizer and 20 bags of Parker's Premium Starter fertilizer are needed to obtain the desired mixture.

Explanation:

Let's assume x represents the number of bags of Vigoro Ultra Turf fertilizer and y represents the number of bags of Parker's Premium Starter fertilizer. We can set up the following equations based on the nutrient content of each bag:

For nitrogen (N): 29x + 18y = 101101

For phosphoric acid (P2O5): 33x + 25y = 103103

For potash (K2O): 44x + 66y = 2828

To solve this system of equations, we can use various methods such as substitution or elimination. Here, we'll use the elimination method:

First, we multiply the first equation by 33, the second equation by 29, and the third equation by 9 to create a common coefficient for x:

957x + 594y = 3339933

957x + 725y = 2988917

396x + 594y = 25452

By subtracting the third equation from the second equation, we obtain:

561x = 2968465

Dividing both sides by 561, we find x = 5285.

Substituting this value back into the first equation, we have:

29(5285) + 18y = 101101

153365 + 18y = 101101

18y = -52264

y = -2904.7

Since the number of bags cannot be negative, we round down to the nearest whole number, resulting in y = 2904.

Therefore, 35 bags of Vigoro Ultra Turf fertilizer and 20 bags of Parker's Premium Starter fertilizer are required to yield a mixture containing 101101 pounds of nitrogen, 103103 pounds of phosphoric acid, and 2828 pounds of potash.

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