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= Introduction to system of linear equation: = System of linear equations is a set of equations thats are contain two or more variables with the same degree of power. It is an essential topic in algebra and has many real-life applications. Solving the system of linear equations helps us to determine the value of variables that satisfy all the given equations. It has various real-world applications such as finding the optimal solution for business problems, determining the location of a point in space, and solving problems in physics, engineering, and many other fields. In this article, we will discuss one such real-life example of the system of linear equations and how it is used in practice.

= Example: A Furniture Manufacturing Company = Suppose a furniture manufacturing company produces two types of chairs — chair A and chair B. The company has two factories — Factory 1 and Factory 2. Factory 1 produces both types of chairs, while Factory 2 produces only chair B. The cost of producing each type of chair is different in each factory. The company has a total budget of $50,000 for the production of these chairs. The company wants to maximize the profit earned by producing and selling these chairs. The profit earned by selling one unit of chair A is $200, and for chair B, it is $150. Using this information, we can form a system of linear equations that can help us find the number of chairs of each type that should be produced in each factory to maximize profit.

Let x and y be the number of chairs produced in Factory 1 for chair A and chair B, respectively, and let z be the number of chairs produced in Factory 2 for chair B. Then, we can write the following system of equations:

Equation 1: x + y = total number of chairs produced in Factory 1

Equation 2: y + z = total number of chairs produced in Factory 2

Equation 3: 100x + 80y + 90z <= 50,000 (total cost of production)

Equation 1 represents the total number of chairs produced in Factory 1, which is the sum of chairs A and B produced in the factory. Similarly, Equation 2 represents the total number of chairs produced in Factory 2, which is the sum of chairs B produced in the factory. Equation 3 represents the total cost of production, which should be less than or equal to the total budget of $50,000.

Now, let’s solve this system of equations to find the number of chairs that should be produced in each factory to maximize profit.

Solving the System of Linear Equations

To solve this system of equations, we can use the method of substitution. Let’s solve Equation 1 for y and Equation 2 for z:

Equation 1: y = -x + total number of chairs produced in Factory 1

Equation 2: z = -y + total number of chairs produced in Factory 2

Substituting these equations in Equation 3, we get:

100x + 80(-x + total number of chairs produced in Factory 1) + 90(-y + total number of chairs produced in Factory 2) <= 50,000

Simplifying this equation, we get:

20x + 8total number of chairs produced in Factory 1 + 9total number of chairs produced in Factory 2 <= 1250

Now, we need to maximize the profit earned by producing and selling these chairs. The profit earned by selling one unit of chair A is $200, and for chair B, it is $150. Therefore, the total profit earned can be represented as:

Total Profit = 200x + 150y + 150z

Substituting y and z in terms of x from Equations