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Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations Elementary Mathematical Modeling Chapter 3. Straight Lines and Linear Functions 3.5. Systems of Linear Equations Da Zheng University of Houston March 5, 2014 Introduction Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations Remember in chapter 2, we learned how to solve the intersection of two functions, that is, by applying the crossing graph method. When it comes to the case of solving a system of linear equations, indeed, we will see that we can also use this method. Before we learn the methods to solve systems of linear equations, let’s look at the definition of a system of linear equations. Introduction Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations A system of linear equations is pair of linear equations of the following form: a1 x + b 1 y = c 1 a2 x + b 2 y = c 2 . We put the two equations together to mean that we are solving for x and y which satisfy the two equations simultaneously. So, to obtain the solutions, we have to make use of both equations. Algebraic Way of Solving Linear Systems of Equations Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations First, we introduce a way to solve by hand. To explain it, let’s see the following example: Example You have $36 to spend on refreshments for a party. Large bags of chips cost $2.00 and drinks cost $0.50. You need to buy five times as many drinks as bags of chips. How many bags of chips and how many drinks can you buy? Algebraic Way of Solving Linear Systems of Equations Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations Let’s say that we buy x bags of chips and y drinks. So the total cost will be 2x + 0.5y. Since we have $36 to spend, 2x + 0.5y = 36. However, we need to buy five times as many drinks as bags of chips. That is, y = 5x. Hence, we obtain the following system of linear equations: 2x + 0.5y = 36 (1) y = 5x (2) Algebraic Way of Solving Linear Systems of Equations Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations To solve it, we first consider equation (2). It tells us y = 4x. So, replacing the variable y in equation (2) by y = 4x, we have 2x + 0.5 · 5x = 36. This is merely a usual linear equation. Solve it, we have 4.5x = 36 =⇒ x = 8. Now, use equation (2), i.e. y = 5x, we obtain y = 40. In-class Problems Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations Problem Solve the following system of linear equations 3x + 2y = 14 2x − y = 0 Graphical Way of Solving Linear Systems of Equations Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations We try solve the following system of linear equations by using another method. 2x + 0.5y = 36 (1) y = 5x (2) Recall the technique of reversing the roles of variables, apply this technique to equation (1), we have y = 72 − 4x. Graphical Way of Solving Linear Systems of Equations Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations But equation (2) tells us we must also have y = 5x. So we are indeed solving the intersection of the two functions: y = 5x and y = 72 − 4x. Now, apply the crossing-graphs method in chapter 2, we have x = 8, y = 40. In-class Problems Elementary Mathematical Modeling Da Zheng 3.5 Systems of Linear Equations Problem Solve the following system of linear equations 3x + 2y = 6 4x − 3y = 8