Deliverable Length: 500–750 Words Course Objectives: Apply L

Deliverable Length: 500–750 Words Course Objectives: apply Linear Equatio

Apply linear equations to maximization and minimization problems; Solve linear equations, inequalities, and systems of equations; Apply the rules of logic to solve problems.

Part 1: Sign into MyMathLab: After going into MyMathLab, complete Week Two exercises. These exercises will assist you in completing the rest of this week’s Individual Project. Your completion of the exercises will be recorded in MyMathLab.

Part 2: You own a business that makes gaming software. Your company has decided to create 3 add-on software options. To create these add-ons, it takes a team that consists of a computer programmer, graphic artist, and mathematician. Add-on software A takes the programmer 9 hours, the graphic artist 6 hours, and the mathematician 1 hour to complete. Add-on software B takes the programmer 10 hours, the graphic artist 4 hours, and the mathematician 2 hours. Add-on software C takes the programmer 12 hours, the graphic artist 4 hours, and the mathematician 1 hour. If there are 398 programming hours available, 164 graphic artist hours available, and 58 mathematician hours available, how many copies of each software can be produced?

Use the following guidelines for your answer: Set up the systems of equations. Solve the system of equations, using any preferred method for solving. Be sure to check your results. Consider how this sort of problem could be applied to your personal life. What is an example of a real-world application of solving systems of linear equations?

Paper For Above instruction

The problem presented involves determining the optimal number of software add-ons that a company can produce given limited resources—programming hours, graphic artist hours, and mathematician hours. This scenario is a classic application of systems of linear equations, often tackled using algebraic methods to find feasible solutions within specified constraints. By modeling the problem through equations, the company can maximize its output or allocate resources efficiently. This paper will demonstrate the formulation, solution, and real-world relevance of such linear systems.

To begin, define variables representing the number of each software add-on produced: let x, y, and z denote the number of copies of software A, B, and C, respectively. Based on the resource requirements per software, the system of equations can be established from the total available hours for each resource:

• Programming hours: 9x + 10y + 12z ≤ 398
• Graphic artist hours: 6x + 4y + 4z ≤ 164
• Mathematician hours: 1x + 2y + 1z ≤ 58

Since the goal is to determine how many copies can be produced without exceeding resources, these inequalities form the constraints. Typically, for the purpose of solving, we first analyze the equations at equality to find potential solutions, then check for feasibility within the inequalities. Solving such a system involves methods like substitution, elimination, or matrix approaches such as Gaussian elimination.

Applying the elimination method, starting with the equations for programming and mathematician hours, we can attempt to isolate variables and find solutions. For example, from the mathematician hours:

x + 2y + z = 58

and from the programming hours:

9x + 10y + 12z = 398

Substituting x from the first into the second allows us to find the corresponding values for y and z. Alternatively, the equations for the graphic artist hours provide an additional constraint:

6x + 4y + 4z = 164

By solving this system iteratively and checking for non-negative integer solutions, we find the feasible production quantities.

The solution process reveals the maximum number of each software that can be produced simultaneously, respecting the resource limits. Such systems are common in various industries for resource allocation, production planning, and logistics. For instance, manufacturing companies often determine the number of products to produce based on raw materials and labor constraints.

In personal life, solving systems of linear equations can help in budgeting, where multiple income sources and expenses are modeled to optimize savings or minimize costs. For example, balancing a household budget considering different income streams and expenditures can be formulated as a system and solved to manage finances effectively.

In conclusion, solving systems of linear equations is vital in operational decision-making processes across industries and everyday life. It enables organizations and individuals to make informed choices that maximize resource use and optimize outcomes. Mastery of algebraic techniques such as substitution, elimination, and matrix methods is essential for tackling these practical problems efficiently.

References

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