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Section 2.1 Lecture Notes MATH 141-501 Linear Systems: There are many times where we need to solve systems of linear equations: Example: Solve 2x + 3y = 7 3x − 15y = 1.5 If the system has two variables, this is really the same as: 1 There are three possibilities. Only one solution: 2 No solution: 3 Infinitely many solutions: 4 Example: One Solution Solve the system x + 51y = 60 −x + y = 980 5 Example: Infinitely Many Solutions Solve the system 5x + 10y = 1 1 1 y =− x+ 2 10 6 No solution: 3x − y = 10 9x − 3y = 115 7 More Than Two Variables What about three variables? Can we solve a system like 2x + 3y − 7z = 1 x + y + z = 121 4x − 5.5y + 13z = 3.4? 8 Geometric Interpretation of A System with Three Variables In the case of three variables, we are dealing with instead of lines. The equation of a Cz = D. is Ax+By+ 9 Systems of Three Variables How many solutions can a system of linear equations with three variables have? Figure 1: Image from Finite Mathematics for the Managerial, Life, and Social Sciences, Tan, 11e 10 Who cares? We do! In terms of real-world applications of math, solving systems of linear equations is one of the biggest. The systems are generally It is not uncommon for businesses to need to solve systems involving variables 11 Example: A music hall has 1800 seats. They offer three ticket prices: child($5), adult($15), and senior($12.50). At a certain concert, the number of children and seniors combined was twice the number of adults. The performance was sold out, and revenue from ticket sales was $23,250. How many tickets of each type did the music hall sell? 12