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. 350 units of sand 445 units of loam 345 units of peat moss

Short answer

Appropriate methods to solve the system of equations will lead to the correct numbers of bags of the three types of potting soil.

Step-by-step solution

Define the Variables

Let's denote the number of bags of potting soil for seedlings as \(x\), for general potting as \(y\) and for hardwood plants as \(z\).

Set Up the Equations

The requirement for each raw material results in an equation. \\ For sand, we get the equation: \(2x + y + 2z = 350\). \\ For loam, we get the equation: \(x + 2y + 2z = 445\). \\ And for peat moss we get the equation: \(x + y + 2z = 345\).

Solve the System of Equations

We have a system of three equations, and three unknowns (x, y, z). We can solve this system by substitution or elimination. This might require some algebraic manipulations of the equations before being able to solve for the variables. In this case, the easiest approach would be to subtract the third equation from the second to eliminate \(z\), then solve the remaining two equations for \(x\) and \(y\). Once \(x\) and \(y\) are found, substitute them into an equation to find \(z\).

Validate the Solutions

Once the solutions have been found, verify these by substituting the values in all original equations to ensure they hold true.

Key concepts

Linear Algebra

Linear algebra is at the heart of a wide variety of calculations in science and engineering. In our context, it provides a framework to describe and solve systems of linear equations — a fundamental aspect when dealing with multiple constraints, as seen in the potting soil company example. The company's problem of producing different types of soil bags can be depicted using variables and linear equations, representing the relationship between the resources available (sand, loam, and peat moss) and the products to be manufactured (different types of potting soil).
Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. Linear algebra allows us to apply methods such as substitution and elimination to find the values of unknowns that meet all requirements, called a 'solution set'. In this scenario, solving the linear equations helps the company determine the exact number of each type of potting soil bag that can be produced with the available materials.

Mathematical Modeling

Mathematical modeling uses mathematical language to describe real-world scenarios. In our exercise, the company's production constraints are modeled using a system of linear equations. This model simplifies the complexities of real-life resources into clear, concise mathematical statements.
Through mathematical modeling, we transform the practical problem of allocating resources (units of sand, loam, and peat moss) into a mathematical form where each type of potting soil and its resource requirements are represented by algebraic expressions. The goal is to find a solution that fits within the constraints outlined—essentially, the most efficient way to use the available resources to meet production goals.

Identifying Constraints and Variables

In this case, each resource's availability becomes a 'constraint,' and the number of bags for each soil type becomes a 'variable.' The mathematical model here aids in making informed decisions by providing a clear view of the resource management required to maximize production while adhering to material limitations.

Systems of Equations Application

The application of systems of equations is a practical approach to finding solutions to problems with several constraints or conditions that must be met simultaneously. Like our potting soil company example, businesses often use systems of equations to plan and optimize their operations. They can determine the mix of products that maximizes the use of available materials while satisfying customer demand.
Application in Business: In our example, the company can only produce as many bags as the limited resources allow. This type of problem, where resources must be allocated efficiently across different products, is ubiquitous in the business world, especially in production and manufacturing sectors.

Optimization and Decision Making

Systems of equations help make optimal decisions by providing a precise mathematical foundation. Knowing how to set up and solve these systems is not just an academic exercise but a real-world skill that can lead to more effective and efficient business operations.

Algebraic Manipulations

Algebraic manipulations are techniques used to transform mathematical expressions and equations into simpler or more useful forms. In solving systems of equations, algebraic manipulations are crucial for isolating variables and reducing complexity. This can include operations like adding, subtracting, multiplying, or dividing both sides of an equation by a number, expanding expressions, factoring, and combining like terms.
In the context of the problem provided, such manipulations allow us to convert a system of equations into simpler equations that are easier to solve. For instance, subtracting the peat moss equation from the loam equation eliminated the variable z, revealing relationships between x and y that we could exploit to find a solution.

Systematic Approach for Solution

A careful, step-by-step approach to algebraic manipulation not only simplifies complex equations but also minimizes errors in calculation, leading us closer to the correct solution in a systematic way. Adequate manipulation skills can make the difference between finding a viable solution or getting lost in calculation.