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# Why It Matters: Systems of Equations and Inequalities

After years of saving, you have finally done it—you rented a space to open your very own coffee shop.  You’ve already painted the walls and set up the furniture.  The next step is to plan your menu.  In considering both taste and cost, you have developed a coffee flavor that is sure to bring customers coming back for more. This soon-to-be signature flavor is made from a mixture of coffee beans.  The first type of coffee bean sells for $8 per pound and the other sells for$14 per pound. You will combine the coffee beans in batches of 100 pounds, and sell them for $9.80 per pound. Now all you need to figure out is how much of each type of coffee bean should you mix together. You can use an equation to relate the amount of each type of coffee bean. If you set the number of pounds of the first coffee bean equal to $x$ and the number of pounds of the first coffee bean equal to $y$, you know that the total is 100. $x+y=100$ You can also relate the cost of each type of bean to the cost of the mixture. The cost of the first type of coffee bean is the number of pounds times the cost per pound, $8x$. Similarly, cost of the second type of coffee bean is the number of pounds times its cost per pound, $14x$. And the total cost is$980 for 100 pounds.

$8x+14y=980$

Now you have two equations, but what can you do to solve for the values of $x$ and $y$?   And what information do those values give you? To find the answers to these and other questions, read on in this module.  There you will learn about combinations of equations, called systems of equations, along with different methods of solving them.

### Learning Objectives

Systems of Linear Equations: Two Variables
• Solve systems of equations by graphing, substitution, and addition
• Identify inconsistent systems of equations containing two variables
• Express the solution of a system of dependent equations containing two variables using standard notations

Systems of Nonlinear Equations and Inequalities

• Solve a system of nonlinear equations using substitution or elimination
• Graph a nonlinear inequality
• Graph a system of nonlinear inequalities

Systems of Linear Equations: Three Variables

• Solve systems of three equations in three variables
• Identify inconsistent systems of equations containing three variables
• Express the solution of a system of dependent equations containing three variables using standard notations

Partial Fractions: an Application of Systems

• Decompose  ${\large\frac{P(x)}{Q(x)}}$, where $Q(x)$ has only nonrepeated linear factors
• Decompose ${\large\frac{P(x)}{Q(x)}}$, where $Q(x)$ has repeated linear factors
• Decompose ${\large\frac{P(x)}{Q(x)}}$, where $Q(x)$ has a nonrepeated irreducible quadratic factor
• Decompose ${\large\frac{P(x)}{Q(x)}}$, where $Q(x)$ has a repeated irreducible quadratic factor