Determining the Number of Solutions to a Systems of Linear Equations

Systems of Linear Equations Lesson #2: Determining the Number of Solutions to a System of Linear Equations Exploration Determining the Number of Solutions to a System of Linear Equations As part of a high school work experience course, three students have been placed in three different burger restaurants. Deja has been placed at Burger Shack, Shelly has been placed at Big's Burgers, and John has been placed at The Burger Haven. Detailed below are the first two orders taken by each student and the total cost calculated by the student for each order. Burger Shack For the first order Deja charges $28 for 4 burgers and 2 salads. For the second order she charges $34 for 6 burgers and a salad. Big’s Burgers Shelly charges $18 for a burger and 3 salads, and $54 for 3 burgers and 9 salads. The Burger Haven John charges $32 for 2 burgers and 4 salads, and $42 for 3 burgers and 6 salads. a) Let x dollars be the cost of a burger and y dollars be the cost of a salad. Write a system of equations for each scenario. Burger Shack Big’s Burgers The Burger Haven 4x + 2y = 28 x + 3y = 18 2x + 4y = 32 6x + y = 34 3x + 9y = 54 3x + 6y = 42 b) Consider the equations for the Burger Shack. Write each equation in terms of y and use a graphing calculator to determine the cost of a burger and the cost of a salad. 4x+2y = 28 6x + y = 34 2y = -4x+28 y = -6x+34 burger $5 y = -2x + 14 salad $4 [Graph of two intersecting lines] c) Repeat part b) for Big’s Burgers. Can you determine the cost of a burger and the cost of a salad? Explain. x + 3y = 18 3x + 9y = 54 3y = -x + 18 9y = -3x + 54 Both equations are the same. ↳ cannot determine the cost. y = -1/3x + 6 y = -1/3x + 6 [Graph of one line, two equations on top of each other] d) Repeat part b) for The Burger Haven. Can you determine the cost of a burger and the cost of a salad? Explain how you can tell that the student must have made an error in at least one of the calculations. 2x + 4y = 32 3x + 6y = 42 4y = -2x + 32 6y = -3x + 42 (1 burger and 2 salads cannot cost $16 and $14) y = -1/2x + 8 y = -1/2x + 7 Both equations have the same slope ∴ lines are parallel [Graph of two parallel lines] Number of Solutions to a System of Equations In all of the examples in Lesson #1, each system of equations had one unique solution. However, we have seen on the previous page that a system of two linear equations may have no solution, only one solution or an infinite number of solutions. Graph each system of equations on the grid provided. State the number of solutions for each system. a) y = x + 5 y = -x + 1 [Graph of two intersecting lines] 1 solution b) y = 2x - 8 y = 2x + 6 [Graph of two parallel lines] no solution c) y = 3x - 2 6x - 2y - 4 = 0 same equation 6x - 4 = 2y y = 3x -2 [Graph of one line, two equations on top of each other] Infinite number of solutions The number of solutions can be determined from the graph as above, or directly from the equations if they are expressed in slope y-intercept form. d) Complete the following chart. Number of Solutions one solution no solution infinitely many Graphical Example [Graph of two intersecting lines] [Graph of two parallel lines] [Graph of one line, two equations on top of each other] Slopes and Intercepts slopes are different slopes are equal and intercepts are different slopes are equal and intercepts are equal Class Ex. #2 Without graphing, analyze each system to determine whether the system has one solution, no solution, or infinitely many solutions. a) 3x + 5y = 15, y = (-3/5)x + 3 5y = -3x + 15 y = (-3/5)x + 3 slopes are the same! no solution b) x - 4y + 8 = 0, y = 1/4x + 2 x + 8 = 4y y = 1/4x + 2 *equations are the same!* *infinitely many solutions* c) 7x + y = 12, x - 6y = 5 y = -7x + 12 -6y = -x + 5 y = 1/6x - 5/6 slopes are different! one solution Complete Assignment Questions #1 - #5