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A system of equations consists of two or more equations with the same set of variables. In this case, we have two linear equations involving two variables: the amount of two different types of coffee. These equations are tasked with finding the quantities needed for a blend. The first equation represents the total weight of the blend, while the second represents the cost per pound of the blend. Together, these equations form a system that can be used to solve for the unknowns once we apply suitable methods like substitution or elimination.

Solving systems of equations is a foundational skill in algebra and is used to model real-world situations. These can include scenarios in business, physics, and engineering. By solving such systems, you can find values that satisfy all given conditions simultaneously, making it a powerful mathematical tool.

The substitution method is one strategy for solving a system of equations, where you solve one of the equations for one variable and substitute that expression into the other equation. This method can be particularly useful when one equation is easily solved for a single variable.

In the context of our coffee blend problem, if we used the substitution method, we would first solve the simple equation for one of the variables. For example, from equation 1, we could express \( x \) in terms of \( y \):

• \( x = 100 - y \)

We would then substitute \( 100 - y \) for \( x \) in equation 2 to solve for \( y \), and subsequently find \( x \). This way, substitution reduces the system to a single equation in one variable, making it easier to handle.

The elimination method involves adding or subtracting the equations in a system to eliminate one of the variables, thus simplifying the problem to a single equation in one variable. In this exercise, we applied the elimination method to find the amount of coffee blends.

By multiplying whole equations and aligning coefficients, elimination helps eliminate variables step by step. For our problem, we multiplied the first equation \( x + y = 100 \) by 5, resulting in \( 5x + 5y = 500 \). Subtracting this from the second equation \( 5x + 6y = 560 \) eliminated \( x \), allowing us to find \( y = 60 \). This gave us a straightforward path to solving for the remaining variable. The clarity and direct approach of elimination make it a popular choice for solving systems of linear equations.

Mathematical modeling involves creating mathematical representations of real-world situations to find insights or solutions. By using variables and equations, we create a model to represent a problem like our coffee blending scenario.

In this exercise, variables \( x \) and \( y \) embody the amounts of two coffee types, and the equations depict constraints such as total weight and price per pound of the blend. This abstraction helps us analyze the situation systematically, predicting outcomes and testing different scenarios.

Modeling is indispensable in decision-making processes across different sectors. It establishes connections between theoretical mathematics and practical issues, streamlining complex information into manageable and solvable components.