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College Algebra K/DC Tuesday, 21 April 2015 • OBJECTIVE TSW solve 2 x 2 systems of equations by the methods of (1) substitution and (2) elimination. • COOKOUT: Monday, 18 May 2015 – If you will be attending, bring $3.00 by Wednesday, 13 May 2015. – Let me know if you have any dietary restrictions. Turn in purple Awards Forms to Mrs. Yerkes ASAP!!! 1 5.1 Systems of Linear Equations Linear Systems ▪ Substitution Method ▪ Elimination Method ▪ Special Systems 5-2 Linear System of Equations A set of first-degree equations in n unknowns is called a linear system of equations. The solutions of a system of equations must satisfy (make Know these true) every equation in the system. vocabulary descriptions! There are three possibilities for the solutions of a linear system of two equations and two variables: 1. One solution 2. No solutions 3. Infinitely many solutions (consistent, (inconsistent system) (consistent, independent system) dependent system) 5-3 Algebraic Methods to Solve Systems Substitution Method. • Use one of the equations to find an expression for one variable in terms of the other Elimination Method. • Use multiplication and/or addition to eliminate a variable from one equation. 5-4 Solving a System by Substitution Solve the system. Solve equation (2) for x: x = 1 + 2y Replace x in equation (1) with 1 + 2y, then solve for y: Distributive property Replace y in equation (2) with 2, then solve for x: 5-5 Solving a System by Substitution The solution of the system is (5, 2). Check this solution in both equations (1) and (2). Solution set: {(5, 2)} 5-6 Solving a System by Substitution To solve the system graphically, solve both equations for y: Graph both Y1 and Y2 in the standard window to find that their point of intersection is (5, 2). 5-7 Solving a System by Elimination Solve the system. Multiply both sides of equation (1) by 2, and then multiply both sides of equation (2) by 3. Add equations (3) and (4), then solve for x. 5-8 Solving a System by Elimination Substitute 4 for x equation (1), then solve for y. The solution of the system is (4, –3). Check this solution in both equations (1) and (2). 5-9 Solving a System by Elimination Solution set: {(4, –3)} 5-10 Solving a System by Elimination The graph confirms that the solution set is {(4, –3)}. 5-11 Solving an Inconsistent System Solve the system. Multiply both sides of equation (1) by 2, then add the resulting equation to equation (2). 14 x 6 y 10 14 x 6 y 10 (3) (2) 0 20 False The system is inconsistent. Solution set: ø 5-12 Solving an Inconsistent System The graphs of the equations are parallel and never intersect. 5-13 Solving a System with Infinitely Many Solutions Solve the system. Multiply both sides of equation (2) by 3, then add the resulting equation to equation (1). The result indicates that the equations of the original system are equivalent. Any ordered pair that satisfies either equation will satisfy the system. 5-14 Solving a System with Infinitely Many Solutions From equation (2), we have The solution set (with x arbitrary) is {(x, 3x – 8)}. From equation (2), we have The solution set (with y arbitrary) is 5-15 Solving a System with Infinitely Many Solutions The graphs of the two equations coincide. 5-16 Assignment • Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd) – Write the system and solve. Use solution sets. – Use the indicated method to solve. – When a system has an infinite number of solutions, use y as the arbitrary variable. • Solve in terms of y. – Due tomorrow, Wednesday, 22 April 2015. 5-17 Assignment: Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd) Due tomorrow, Wednesday, 22 April 2015. For 7-17, solve by substitution. 4 x 3 y 13 7) 9) x y 5 x 5y 8 x 6y 11) 8 x 10y 22 3x y 6 7 x y 10 3 y x 10 13) 2 x 6 y 18 17) 29 5 y 3 x For 19-27, solve by elimination. 19) 3 x y 4 21) 15) x 3y 12 23) 5 x 7y 6 10 x 3 y 46 27) 3y 5 x 6 xy 2 2 x 3 y 7 5 x 4 y 17 25) 6 x 7 y 2 0 x y 4 7 x 6 y 26 0 2 3 3 x 3y 15 2 2 5-18 Assignment: Sec. 5.1: pp. 504-505 (7-27 odd, 31-39 odd) Due tomorrow, Wednesday, 22 April 2015. Solve each system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with y arbitrary. 31) 9 x 5 y 1 18 x 10 y 1 3 3) 4 x y 9 8 x 2y 18 35) 5 x 5 y 3 0 x y 12 0 37) 7 x 2y 6 14 x 4 y 12 39) Get from the book. 5-19