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Algebra 2 Section 1-9

This document provides an example of solving a system of 3 linear equations in 3 variables. It shows setting the equations equal to each other to eliminate variables, resulting in a single variable that can be solved for. Plugging this solution back into the original equations finds the solutions for the other 2 variables, providing the ordered triple solution. The example solves for x = -2, y = 6, z = 4.

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Section 1-9 Solving Systems of Equations in Three Variables
Essential Questions • How do you solve systems of linear equations in three variables? • How do you solve real-world problems using systems of linear equations in three variables?
Vocabulary 1. Ordered Triple:
Vocabulary 1. Ordered Triple: The solution of a system of equations in three variables will have three answers to fit the three variables; must satisfy all three equations
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 2x + y − z = 5 ⎧ ⎨ ⎩⎪ 1 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10 9x + 5y = 12
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10 9x + 5y = 12A
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16 5x + 6y = 26
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16 5x + 6y = 26B
Example 1 Solve the system of equations. 9x + 5y = 12 5x + 6y = 26 ⎧ ⎨ ⎩⎪ A B
Example 1 Solve the system of equations. (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5 y = 6
Example 1 Solve the system of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5 y = 6 5(−2)+ 6(6) = 26
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5 −z = 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5 −z = 3 z = −3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16 −2 + 24 − 6 = 16
Example 1 Solve the system of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16 −2 + 24 − 6 = 16 (x,y ,z ) = (−2,6,−3)
Solving a 3X3 System
Solving a 3X3 System 1. Group equations so you have two sets of two equations
Solving a 3X3 System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set
Solving a 3X3 System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations
Solving a 3X3 System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution)
Solving a 3X3 System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution) 5. Plug back into an original equation to find the third variable
Solving a 3X3 System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution) 5. Plug back into an original equation to find the third variable 6. Check your answer in each of the original equations
Example 2 Solve the system of equations. 2x + y − 3z = 5 x + 2y − 4z = 7 6x + 3y − 9z + 15 ⎧ ⎨ ⎪ ⎩ ⎪
Example 2 Solve the system of equations. 2x + y − 3z = 5 x + 2y − 4z = 7 6x + 3y − 9z + 15 ⎧ ⎨ ⎪ ⎩ ⎪ Infinite Solutions
Example 3 Solve the system of equations. 3x − y − 2z = 4 6x − 2y − 4z = 11 9x − 3y − 6z = 12 ⎧ ⎨ ⎪ ⎩ ⎪
Example 3 Solve the system of equations. 3x − y − 2z = 4 6x − 2y − 4z = 11 9x − 3y − 6z = 12 ⎧ ⎨ ⎪ ⎩ ⎪ No Solutions
Example 4 There are 49,000 seats in Sheckyville Stadium. Tickets for the seats in the upper level sell for $25, the ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number of seats in the middle and bottom levels together equals the number os states in the upper level. If all of the seats are side for an event, the total revenue is $1,419,500. How many seats are there in each level? Does your answer seem reasonable? Explain.
Example 4 x = upper level, y = middle level, z = lower level
Example 4 x + y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level
Example 4 x + y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level (x,y,z ) = (24500, 10100, 14400)
Example 4 x + y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level (x,y,z ) = (24500, 10100, 14400) There are 24,500 upper level seats, 10,100 middle level seats, and 14,400 lower level seats.

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Algebra 2 Section 1-9

  • 1.
    Section 1-9 Solving Systemsof Equations in Three Variables
  • 2.
    Essential Questions • Howdo you solve systems of linear equations in three variables? • How do you solve real-world problems using systems of linear equations in three variables?
  • 3.
  • 4.
    Vocabulary 1. Ordered Triple:The solution of a system of equations in three variables will have three answers to fit the three variables; must satisfy all three equations
  • 5.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪
  • 6.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1
  • 7.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2
  • 8.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
  • 9.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 2x + y − z = 5 ⎧ ⎨ ⎩⎪ 1 2
  • 10.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2
  • 11.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10
  • 12.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10 9x + 5y = 12
  • 13.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 (2x + y − z = 5)(2) ⎧ ⎨ ⎩⎪ 1 2 5x + 3y + 2z = 2 4x + 2y − 2z = 10 9x + 5y = 12A
  • 14.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
  • 15.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3
  • 16.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3
  • 17.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16
  • 18.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16 5x + 6y = 26
  • 19.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 (2x + y − z = 5)(2) x + 4y + 2z = 16 ⎧ ⎨ ⎩⎪ 2 3 4x + 2y − 2z = 10 x + 4y + 2z = 16 5x + 6y = 26B
  • 20.
    Example 1 Solve thesystem of equations. 9x + 5y = 12 5x + 6y = 26 ⎧ ⎨ ⎩⎪ A B
  • 21.
    Example 1 Solve thesystem of equations. (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
  • 22.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
  • 23.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B
  • 24.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29
  • 25.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2
  • 26.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12
  • 27.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12
  • 28.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12
  • 29.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18
  • 30.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30
  • 31.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5
  • 32.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5 y = 6
  • 33.
    Example 1 Solve thesystem of equations. 54x + 30y = 72 −25x − 30y = −130 29x = −58 (9x + 5y = 12)(6) (5x + 6y = 26)(−5) ⎧ ⎨ ⎩⎪ A B 29 29 x = −2 9x + 5y = 12 9(−2)+ 5y = 12 −18 + 5y = 12 +18 +18 5y = 30 5 5 y = 6 5(−2)+ 6(6) = 26
  • 34.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪
  • 35.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1
  • 36.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2
  • 37.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
  • 38.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5
  • 39.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5
  • 40.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5
  • 41.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5
  • 42.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5 −z = 3
  • 43.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 2x + y − z = 5 2(−2)+ 6 − z = 5 −4 + 6 − z = 5 2 − z = 5 −z = 3 z = −3
  • 44.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3
  • 45.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2
  • 46.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2
  • 47.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2
  • 48.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5
  • 49.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5
  • 50.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5
  • 51.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16
  • 52.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16
  • 53.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16 −2 + 24 − 6 = 16
  • 54.
    Example 1 Solve thesystem of equations. 5x + 3y + 2z = 2 2x + y − z = 5 x + 4y + 2z = 16 ⎧ ⎨ ⎪ ⎩ ⎪ 1 2 3 5x + 3y + 2z = 2 5(−2)+ 3(6)+ 2(−3) = 2 −10 + 18 − 6 = 2 2x + y − z = 5 2(−2)+ 6 + 3 = 5 −4 + 9 = 5 x + 4y + 2z = 16 −2 + 4(6)+ 2(−3) = 16 −2 + 24 − 6 = 16 (x,y ,z ) = (−2,6,−3)
  • 55.
  • 56.
    Solving a 3X3System 1. Group equations so you have two sets of two equations
  • 57.
    Solving a 3X3System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set
  • 58.
    Solving a 3X3System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations
  • 59.
    Solving a 3X3System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution)
  • 60.
    Solving a 3X3System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution) 5. Plug back into an original equation to find the third variable
  • 61.
    Solving a 3X3System 1. Group equations so you have two sets of two equations 2. Eliminate the same variable in each set 3. Set up a new system of two equations 4. Solve the 2X2 system (check the solution) 5. Plug back into an original equation to find the third variable 6. Check your answer in each of the original equations
  • 62.
    Example 2 Solve thesystem of equations. 2x + y − 3z = 5 x + 2y − 4z = 7 6x + 3y − 9z + 15 ⎧ ⎨ ⎪ ⎩ ⎪
  • 63.
    Example 2 Solve thesystem of equations. 2x + y − 3z = 5 x + 2y − 4z = 7 6x + 3y − 9z + 15 ⎧ ⎨ ⎪ ⎩ ⎪ Infinite Solutions
  • 64.
    Example 3 Solve thesystem of equations. 3x − y − 2z = 4 6x − 2y − 4z = 11 9x − 3y − 6z = 12 ⎧ ⎨ ⎪ ⎩ ⎪
  • 65.
    Example 3 Solve thesystem of equations. 3x − y − 2z = 4 6x − 2y − 4z = 11 9x − 3y − 6z = 12 ⎧ ⎨ ⎪ ⎩ ⎪ No Solutions
  • 66.
    Example 4 There are49,000 seats in Sheckyville Stadium. Tickets for the seats in the upper level sell for $25, the ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number of seats in the middle and bottom levels together equals the number os states in the upper level. If all of the seats are side for an event, the total revenue is $1,419,500. How many seats are there in each level? Does your answer seem reasonable? Explain.
  • 67.
    Example 4 x =upper level, y = middle level, z = lower level
  • 68.
    Example 4 x +y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level
  • 69.
    Example 4 x +y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level (x,y,z ) = (24500, 10100, 14400)
  • 70.
    Example 4 x +y + z = 49,000 y + z = x 25x + 30y + 35z = 1,419,500 ⎧ ⎨ ⎪ ⎩ ⎪ x = upper level, y = middle level, z = lower level (x,y,z ) = (24500, 10100, 14400) There are 24,500 upper level seats, 10,100 middle level seats, and 14,400 lower level seats.