Read the explanations, follow the examples, take notes, and complete the You Trys. What is a system of linear equations? A system of linear equations consists of two or more linear equations with the same variables. In Sections 4.1–4.3, we will learn how to solve systems of two equations in two variables. Some examples of such systems are 2 x 5y 5 y 1 3 x 8 3x y 1 x 4y 1 5x 6y 10 x 2 In the third system, we see that x 2 is written with only one variable. However, we can think of it as an equation in two variables by writing it as x 0y 2. It is also possible to solve systems of inequalities. In Section 4.5, we will learn how to solve linear inequalities in two variables. 1 Determine Whether an Ordered Pair Is a Solution of a System We will begin our work with systems of equations by determining whether an ordered pair is a solution of the system. Definition A solution of a system of two equations in two variables is an ordered pair that is a solution of each equation in the system. EXAMPLE 1 In-Class Example 1 Determine whether (5, 1) is a solution of each system of equations. a) 6x 7y 14 2x 9y 1 b) x 4y 1 3x 8y 23 Answer: a) not a solution b) solution Determine whether (2, 3) is a solution of each system of equations. a) y x 1 b) 4x 5y 7 x 2y 8 3x y 4 Solution a) If (2, 3) is a solution of y x 1 x 2y 8 then when we substitute 2 for x and 3 for y, the ordered pair will make each equation true. y x 1 x 2 y 8 3 2 1 Substitute. 2 2(3) 8 Substitute. 2 6 8 3 3 True 8 8 True Since (2, 3) is a solution of each equation, it is a solution of the system. b) We will substitute 2 for x and 3 for y to see whether (2, 3) satisfi es (is a solution of) each equation. 4 x 5y 7 3x y 4 4(2) 5(3) 7 Substitute. 3(2) 3 4 Substitute. 8 15 7 6 3 4 7 7 True 9 4 False Although (2, 3) is a solution of the fi rst equation, it does not satisfy 3x y 4. Therefore, (2, 3) is not a solution of the system. When given the ordered pair (2, 3), write down x 2 and y 3 on scratch paper. This may help you remember which value is x, and which is y. www.mhhe.com/messersmith SECTION 4.1 Solving Systems by Graphing 245
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