6-3 Solving Systems Using Elimination (Combination) Hubarth Algebra

Ex 1 Adding Equations Solve by elimination. 2x + 3y = 11 –2x + 9y = 1 Step 1: Eliminate x because the sum of the coefficients is 0. 2x + 3y = 11 –2x + 9y =1 0 + 12y = 12 Addition Property of Equality y = 1 Solve for y. Step 2: Solve for the eliminated variable x using either original equation. 2x + 3y = 11 Choose the first equation. 2x + 3(1) = 11 Substitute 1 for y. 2x + 3 = 11 Solve for x. 2x = 8 x = 4 Since x = 4 and y = 1, the solution is (4, 1). Check: See if (4, 1) makes the equation not used in Step 2 true. –2(4) + 9(1) 1 Substitute 4 for x and 1 for y into the second equation. –8 + 9 1 1 = 1

Ex 2 Application On a special day, tickets for a minor league baseball game cost $5 for adults and $1 for students. The attendance that day was 1139, and $3067 was collected. Write and solve a system of equations to find the number of adults and the number of students that attended the game. Define: Let a = number of adults Let s = number of students Relate: total number at the game total amount collected Write: a + s = 1139 5 a +1 s = 3067 Step 1: Eliminate one variable. a + s = 1139 5a + s = 3067 –4a + 0 = –1928 Subtraction Property of Equality a = 482 Solve for a. Step 2: Solve for the eliminated variable using either of the original equations. a + s = 1139 Choose the first equation. 482 + s = 1139 Substitute 482 for a. s = 657 Solve for s. There were 482 adults and 657 students at the game.

Ex 3 Multiplying One Equation Solve by elimination. 3x + 6y = –6 –5x – 2y = –14 Step 1: Eliminate one variable. Start with the given system. 3x + 6y = –6 –5x – 2y = –14 To prepare to eliminate y, multiply the second equation by 3. 3x + 6y = –6 3(–5x – 2y = –14) Add the equations to eliminate y. 3x + 6y = –6 –15x – 6y = –42 –12x + 0 = –48 Step 2: Solve for x. –12x = –48 x = 4 Step 3: Solve for the eliminated variable using either of the original equations. 3x + 6y = –6 Choose the first equation. 3(4) + 6y = –6 Substitute 4 for x. 12 + 6y = –6 Solve for y. 6y = –18 y = –3 The solution is (4, –3).

Ex 4 Multiplying Both Equations Solve by elimination. 3x + 5y = 10 5x + 7y = 10 Step 1: Eliminate one variable. Start with the given system. 3x + 5y = 10 5x + 7y = 10 To prepare to eliminate x, multiply one equation by 5 and the other equation by 3. 5(3x + 5y = 10) 3(5x + 7y = 10) Subtract the equations to eliminate x. 15x + 25y = 50 15x + 21y = 30 0 + 4y = 20 Step 2: Solve for y. 4y = 20 y = 5 3x + 5y = 10 Use the first equation. 3x + 5(5) = 10 Substitute 5 for y. 3x + 25 = 10 3x = –15 x = –5 Step 3: Solve for the eliminated variable x using either of the original equations. The solution is (–5, 5).

Practice 1. Solve by Elimination x + y = 10 x – y = 8 b. -2x + 15y = -32 7x – 5y = 17 c. 15x + 3y = 9 10x + 7y = -4 (9, 1) (1, -2) (1, -2) 2. Your class sells a total of 64 tickets to a play. A student tickets cost $1, and an adult ticket Costs $2.50. Your class collects $109 in total ticket sales. How many adult tickets did you sell? How many student tickets did you sell? 30 adults, 34 students