Question

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 750 units of loam 450 units of peat moss

Short answer

The company can produce 50 bags of potting soil for seedlings, 300 bags of potting soil for general potting, and 50 bags of potting soil for hardwood plants.

Step-by-step solution

Set up the system of equations

Create three equations based on the amount of each raw material used for each type of potting soil. Let's denote \n\n\( x \) as the number of seedling soil bags,\n\n\( y \) as the number of general potting soil bags,\n\n\( z \) as the number of hardwood soil bags.\n\nForm the systems of equations based on given raw materials:\n\nSand: \( 2x + y + 2z = 500 \)\n\nLoam: \( x + 2y + 2z = 750 \)\n\nPeat Moss: \( x + y + 2z = 450 \)

Solve the system of equations

To solve the system of equations, you could use the elimination or substitution method. But because of the structure of the given system, it is best to subtract the third equation from the second one, which results in:\n\nLoam - Peat Moss: \( (x + 2y + 2z) - (x + y + 2z) = 750 - 450 \)\n\nSolving gives \( y = 300 \). Once we've solved for \( y \), substitute it into the first and third equations:\n\nSubstitute \( y \) in Sand equation: \( 2x + 300 + 2z = 500 \), which simplifies to \( x + z = 100 \)\n\nSubstitute \( y \) in Peat Moss equation: \( x + 300 + 2z = 450 \), which simplifies to \( x + 2z = 150 \)\n\nBy subtracting these two new equations, we find that \( z = 50 \). Substituting \( y = 300 \) and \( z = 50 \) back into any of our original equations allows us to solve for \( x = 50 \).

Interpret the solution

The solution \( (x, y, z) = (50, 300, 50) \) indicates that the company can produce 50 bags of seedling soil, 300 bags of general potting soil, and 50 bags of hardwood soil with the given amounts of raw materials.