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Honors Algebra II 3.5 Solving Systems with Three Variables. 1 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3 z 4 a. 2 x 3 y z 15 4 x 3 y z 19 2 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 3 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. 4 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! 5 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! x 3 y 3z 4 2 x 3 y z 15 6 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 Equation #1 and #2 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! x 3 y 3z 4 2 x 3 y z 15 3x 2z 11 7 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 x 3 y 3z 4 2 x 3 y z 15 3x Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #2 2 x 3 y z 15 + Equation #3 4 x 3 y z 19 2z 11 8 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 x 3 y 3z 4 2 x 3 y z 15 3x Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. ZAP THE y-TERMS!!! Equation #2 2 x 3 y z 15 + Equation #3 4 x 3 y z 19 2z 11 9 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 x 3 y 3z 4 2 x 3 y z 15 3x Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #2 2 x 3 y z 15 + Equation #3 4 x 3 y z 19 2z 11 10 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 x 3 y 3z 4 2 x 3 y z 15 3x 2z 11 Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #2 2 x 3 y z 15 + Equation #3 4 x 3 y z 19 6x 2z 34 11 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 + Equation #2 Equation #1 Equation #2 Equation #3 x 3 y 3z 4 2 x 3 y z 15 3x Equation #2 and #3 have the opposite coefficients for y, so that term can be easily eliminated. Equation #2 2 x 3 y z 15 + Equation #3 4 x 3 y z 19 2z 11 6x 2z 34 Now we have two equations with two variables… just like we’ve been doing! 12 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 13 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!! 14 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 9 x 45 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!! 15 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 9 x 45 The z-values are opposites, so add straight down to eliminate the z-terms. ZAP THE z-TERMS!!! x 5 16 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 The z-values are opposites, so add straight down to eliminate the z-terms. 9 x 45 ZAP THE z-TERMS!!! x 5 We now have one of the three values. Just two more to go!!! 17 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 Equation #1 Equation #2 Equation #3 Equation #1 + #2 Equation #2 + #3 3x 2 z 11 6 x 2 z 34 x 5 18 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 Plug the value for x back into one of the equations and solve for z. 19 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 Plug the value for x back into one of the equations and solve for z. 3x 2 z 11 3(5) 2 z 11 20 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 Plug the value for x back into one of the equations and solve for z. 3 x 2 z 11 3(5) 2 z 11 15 2 z 11 21 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 Equation #1 Equation #2 Equation #3 Equation #1 + #2 3x 2 z 11 Equation #2 + #3 6 x 2 z 34 Plug the value for x back into one of the equations and solve for z. 3 x 2 z 11 3(5) 2 z 11 15 2 z 11 2 z 4 z 2 We now have TWO of the three values. Just ONE more to go!!! 22 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 23 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 2 x 3 y z 15 2(5) 3 y (2) 15 24 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 2 x 3 y z 15 2(5) 3 y (2) 15 10 3 y 2 15 25 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 2 x 3 y z 15 2(5) 3 y (2) 15 10 3 y 2 15 26 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 2 x 3 y z 15 2(5) 3 y (2) 15 10 3 y 2 15 12 3 y 15 3y 3 27 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 Plug both values back into one of the original equations and solve for the final value, y. 2 x 3 y z 15 2(5) 3 y (2) 15 10 3 y 2 15 12 3 y 15 3y 3 y 1 28 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 3 y 3z 4 a. 2 x 3 y z 15 4 x 3 y z 19 x 5 z 2 Equation #1 Equation #2 Equation #3 2 x 3 y z 15 2(5) 3 y (2) 15 10 3 y 2 15 12 3 y 15 WOO HOO!!! We’re done! We’ve found all three values!!! Our answer is (5, 1, -2) 3y 3 y 1 29 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3 x y 2 z 1 x yz 0 30 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #3 31 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 ZAP THE y-TERMS!!! Equation #2 Equation #3 32 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 Equation #2 x y z 0 Equation #3 Equation #1 2 x y z 5 +Equation #2 3x y 2 z 1 Equation #1 ZAP THE y-TERMS!!! 33 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 Equation #2 x y z 0 Equation #3 Equation #1 2 x y z 5 +Equation #2 3x y 2 z 1 5x z4 Equation #1 ZAP THE y-TERMS!!! 34 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 Equation #2 x y z 0 Equation #3 Equation #1 2 x y z 5 +Equation #2 3x y 2 z 1 Equation #1 5x z4 ZAP THE y-TERMS!!! Equation #1 +Equation #3 2 x y z 5 x yz0 35 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 Equation #2 x y z 0 Equation #3 Equation #1 2 x y z 5 +Equation #2 3x y 2 z 1 Equation #1 5x z4 ZAP THE y-TERMS!!! 2 x y z 5 x yz0 +Equation #3 3x 2 z 5 Equation #1 36 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz 0 Equation #1 +Equation #2 Equation #1 Equation #2 Equation #3 2 x y z 5 3x y 2 z 1 5x ZAP THE y-TERMS!!! z4 2 x y z 5 +Equation #3 x y z 0 3x 2 z 5 Equation #1 37 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #3 Equation #1 + #2 5 x z 4 Equation #1 + #3 3x 2 z 5 38 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 ZAP THE z-TERMS!!! Equation #2 Equation #3 Equation #1 + #2(5 x z 4) 2 Equation #1 + #33x 2 z 5 10 x 2 z 8 3x 2 z 5 39 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 ZAP THE z-TERMS!!! Equation #2 Equation #3 Equation #1 + #2(5 x z 4) 2 Equation #1 + #33x 2 z 5 10 x 2 z 8 3x 2 z 5 13x 13 x 1 40 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #3 Equation #1 + #2(5 x z 4) 2 Equation #1 + #33x 2 z 5 10 x 2 z 8 3x 2 z 5 13x 13 x 1 One down, two to go! We have our first value, x = 1. 41 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #1 + #2 5 x z 4 Equation #1 + #3 3x 2 z 5 Equation #3 x 1 42 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 x 1 Equation #1 + #2 5 x z 4 Equation #1 Equation #1 + #3 3x 2 z 5 Equation #2 Equation #3 5x z 4 5(1) z 4 43 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 x 1 Equation #1 + #2 5 x z 4 Equation #1 Equation #1 + #3 3x 2 z 5 Equation #2 Equation #3 5x z 4 5(1) z 4 5 z 4 z 1 44 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 x 1 Equation #1 Equation #2 Equation #3 5x z 4 5(1) z 4 5 z 4 z 1 Two down! We’re almost there! 45 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #3 x 1 z 1 46 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 x 1 z 1 Equation #1 Equation #2 Equation #3 2x y z 5 2(1) y (1) 5 47 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 x 1 z 1 Equation #1 Equation #2 Equation #3 2x y z 5 2(1) y (1) 5 2 y 1 5 3 y 5 y2 48 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2 x y z 5 b. 3x y 2 z 1 x yz0 Equation #1 Equation #2 Equation #3 x 1 z 1 2x y z 5 2(1) y (1) 5 2 y 1 5 3 y 5 Woo Hoo! We’re Done! (1, 2, -1) y2 49 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2x y z 5 c. x 4 y 2 z 16 15 x 6 y 2 z 12 50 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. 2x y z 5 c. x 4 y 2 z 16 15 x 6 y 2 z 12 The solution is (-2, 6, -3) 51 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 4 y 5 z 7 d. 3 x 2 y 3 z 7 2 x y 5z 8 52 Solving Systems of Equations in 3 variables by Elimination: Example 1: Solve the system by elimination. (Number the equations to make it easier!) GOAL: Eliminate one of the variables at a time. x 4 y 5 z 7 d. 3x 2 y 3z 7 2 x y 5z 8 The solution is (2, -1, 1) 53 Solving Systems of Equations in 3 variables by Substitution: GOAL: Substitute for one of the variables to create a system with 2 variables!!! x 2 y z 4 e. 4 x y 2 z 1 2 x 2 y z 10 54 Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!! x 2 y z 4 e. 4 x y 2 z 1 2 x 2 y z 10 The solution is (2, 1, -4) 55 Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!! x 3 y z 6 f. 2 x 5 y z 2 x y 2 z 7 56 Solving Systems of Equations in 3 variables by Substitution: Example 3: Solve the system by substitution. (Number the equations to make it easier!) GOAL: Substitute for one of the variables to create a system with 2 variables!!! x 3 y z 6 f. 2 x 5 y z 2 x y 2 z 7 The solution is (4, 1, 5) 57
