760 CHAPTER 11 Systems of Equations and Inequalities 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 sold 525 tickets and had revenue of $4630. How many of each type of ticket were sold? Solution EXAMPLE 6 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, then the given information results in the system of equations + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 10 8 4630 525 (1) (2) Using the method of elimination, first multiply equation (2) by −8, and then add the equations. x y x y 10 8 4630 8 8 4200 + = − − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x x 2 430 215 = = Since + = x y 525, then = − = − = y x 525 525 215 310. So 215 nondiscounted tickets and 310 senior discount tickets were sold. Multiply equation (2) by −8. Add the equations. 3 Identify Inconsistent Systems of Equations Containing Two Variables The previous examples dealt with consistent systems of equations that had a single solution. The next two examples deal with two other possibilities that may occur, the first being a system that has no solution. Identifying an Inconsistent System of Linear Equations Solve: + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 5 4 2 8 (1) (2) Solution EXAMPLE 7 We choose to use the method of substitution and solve equation (1) for y. x y y x 2 5 2 5 + = = − + Now substitute − +x2 5 for y in equation (2) and solve for x. x y x x x x 4 2 8 4 2 2 5 8 4 4 10 8 10 8 ( ) + = + − + = − + = = This statement is false.We conclude that the system has no solution and is inconsistent. (1) Subtract x2 from both sides. (2) Substitute y x2 5. =− + Multiply out. Simplify. Figure 4 illustrates the pair of lines whose equations form the system in Example 7. Notice that the graphs of the two equations are lines, each with slope −2; one has a y-intercept of 5, the other a y-intercept of 4. The lines are parallel and have no point of intersection.This geometric statement is equivalent to the algebraic statement that the system has no solution. 4 Express the Solution of a System of Dependent Equations Containing Two Variables Solving a System of Dependent Equations Solve: + = − − =− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y 2 4 6 3 12 (1) (2) EXAMPLE 8 Figure 4 8 Y1 5 22x 1 5 Y2 5 22x 1 4 22 28 8
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