SECTION 11.1 Systems of Linear Equations: Substitution and Elimination 755 11.1 Systems of Linear Equations: Substitution and Elimination Now Work the ‘Are You Prepared?’ problems on page 765. • Linear Equations (Section A.6, pp. A45–A47) • Lines (Section 1.5, pp. 32–43) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Solve Systems of Equations by Substitution (p. 758) 2 Solve Systems of Equations by Elimination (p. 758) 3 Identify Inconsistent Systems of Equations Containing Two Variables (p. 760) 4 Express the Solution of a System of Dependent Equations Containing Two Variables (p. 760) 5 Solve Systems of Three Equations Containing Three Variables (p. 761) 6 Identify Inconsistent Systems of Equations Containing Three Variables (p. 763) 7 Express the Solution of a System of Dependent Equations Containing Three Variables (p. 763) Movie Theater Ticket Sales A movie theater sells tickets for $10.00 each, with seniors receiving a discount of $2.00. One evening the theater had $4630 in revenue. If x represents the number of tickets sold at $10.00 and y the number of tickets sold at the discounted price of $8.00, write an equation that relates these variables. EXAMPLE 1 Solution Each nondiscounted ticket costs $10.00, so x tickets bring in x 10 dollars. Similarly, y discounted tickets bring in y8 dollars. Because the total revenue is $4630, we must have + = x y 10 8 4630 Throughout the text we have used different types of equations as models when solving applications. In this chapter we learn about systems of equations and their use in applications. In Example 1, suppose that we also know that 525 tickets were sold that evening. Then we have another equation relating the variables x and y: + = x y 525 The two equations + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 10 8 4630 525 form a system of equations. In general, a system of equations is a collection of two or more equations, each containing one or more variables. Example 2 gives some illustrations of systems of equations. Tim Berners-Lee (1955-Present) Credit: dpa picture alliance/Alamy Stock Photo In 1989, Tim Berners-Lee was working at CERN, the European Organization for Nuclear Research. At CERN, Tim developed a project called WorldWideWeb (WWW). While implementing the project, the scientist and his assistant developed the URI identifier, HTTP protocol and HTML markup language. His work led to the way we now interact with information on the internet. Examples of Systems of Equations (a) + = − + =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 5 4 6 2 (b) + = + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y 5 2 4 2 EXAMPLE 2 (1) Two equations containing two variables, x and y (2) (1) Two equations containing two variables, x and y (2) (continued)
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