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3.4 Solving Systems of Equations in Three Variables

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3.4 Solving Systems of Equations in Three Variables Algebra II Mrs. Aguirre Fall 2013 Objective Solve a system of equations in three variables. Application Courtney ... – PowerPoint PPT presentation

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Title: 3.4 Solving Systems of Equations in Three Variables


1
3.4 Solving Systems of Equations in Three
Variables
  • Algebra II
  • Mrs. Aguirre
  • Fall 2013

2
Objective
  • Solve a system of equations in three variables.

3
Application
  • Courtney has a total of 256 points on three
    Algebra tests. Her score on the first test
    exceeds his score on the second by 6 points. Her
    total score before taking the third test was 164
    points. What were Courtneys test scores on the
    three tests?

4
Explore
  • Problems like this one can be solved using a
    system of equations in three variables. Solving
    these systems is very similar to solving systems
    of equations in two variables. Try solving the
    problem
  • Let f Courtneys score on the first test
  • Let s Courtneys score on the second test
  • Let t Courtneys score on the third test.

5
Plan
  • Write the system of equations from the
    information given.
  • f s t 256
  • f s 6
  • f s 164

The total of the scores is 256.
The difference between the 1st and 2nd is 6
points.
The total before taking the third test is the sum
of the first and second tests..
6
Solve
  • Now solve. First use elimination on the last two
    equations to solve for f.
  • f s 6
  • f s 164
  • 2f 170
  • f 85

The first test score is 85.
7
Solve
  • Then substitute 85 for f in one of the original
    equations to solve for s.
  • f s 164
  • 85 s 164
  • s 79

The second test score is 79.
8
Solve
  • Next substitute 85 for f and 79 for s in f s
    t 256.
  • f s t 256
  • 85 79 t 256
  • 164 t 256
  • t 92

The third test score is 92.
Courtneys test scores were 85, 79, and 92.
9
Examine
  • Now check your results against the original
    problem.
  • Is the total number of points on the three tests
    256 points?
  • 85 79 92 256 ?
  • Is one test score 6 more than another test score?
  • 79 6 85 ?
  • Do two of the tests total 164 points?
  • 85 79 164 ?
  • Our answers are correct.

10
Solutions?
  • You know that a system of two linear equations
    doesnt necessarily have a solution that is a
    unique ordered pair. Similarly, a system of
    three linear equations in three variables doesnt
    always have a solution that is a unique ordered
    triple.

11
Graphs
  • The graph of each equation in a system of three
    linear equations in three variables is a plane.
    Depending on the constraints involved, one of the
    following possibilities occurs.

12
Graphs
  1. The three planes intersect at one point. So the
    system has a unique solution.
  • 2. The three planes intersect in a line. There
    are an infinite number of solutions to the
    system.

13
Graphs
  • 3. Each of the diagrams below shows three planes
    that have no points in common. These systems of
    equations have no solutions.

14
Ex. 1 Solve this system of equations
  • Substitute 4 for z and 1 for y in the first
    equation, x 2y z 9 to find x.
  • x 2y z 9
  • x 2(1) 4 9
  • x 6 9
  • x 3 Solution is (3, 1, 4)
  • Check
  • 1st 3 2(1) 4 9 ?
  • 2nd 3(1) -4 1 ?
  • 3rd 3(4) 12 ?
  • Solve the third equation, 3z 12
  • 3z 12
  • z 4
  • Substitute 4 for z in the second equation 3y z
    -1 to find y.
  • 3y (4) -1
  • 3y 3
  • y 1

15
Ex. 2 Solve this system of equations
  • Set the next two equations together and multiply
    the first times 2.
  • 2(x 3y 2z 11)
  • 2x 6y 4z 22
  • 3x - 2y 4z 1
  • 5x 4y 23
  • Next take the two equations that only have x and
    y in them and put them together. Multiply the
    first times -1 to change the signs.
  • Set the first two equations together and multiply
    the first times 2.
  • 2(2x y z 3)
  • 4x 2y 2z 6
  • x 3y -2z 11
  • 5x y 17

16
Ex. 2 Solve this system of equations
  • Now you have y 2. Substitute y into one of the
    equations that only has an x and y in it.
  • 5x y 17
  • 5x 2 17
  • 5x 15
  • x 3
  • Now you have x and y. Substitute values back
    into one of the equations that you started with.
  • 2x y z 3
  • 2(3) - 2 z 3
  • 6 2 z 3
  • 4 z 3
  • z -1
  • Next take the two equations that only have x and
    y in them and put them together. Multiply the
    first times -1 to change the signs.
  • -1(5x y 17)
  • -5x - y -17
  • 5x 4y 23
  • 3y 6
  • y 2

17
Ex. 2 Check your work!!!
  • Solution is (3, 2, -1)
  • Check
  • 1st 2x y z
  • 2(3) 2 1 3 ?
  • 2nd x 3y 2z 11
  • 3 3(2) -2(-1) 11 ?
  • 3rd 3x 2y 4z
  • 3(3) 2(2) 4(-1) 1 ?

18
Ex. 2 Solve this system of equations
  • Now you have y 2. Substitute y into one of the
    equations that only has an x and y in it.
  • 5x y 17
  • 5x 2 17
  • 5x 15
  • x 3
  • Now you have x and y. Substitute values back
    into one of the equations that you started with.
  • 2x y z 3
  • 2(3) - 2 z 3
  • 6 2 z 3
  • 4 z 3
  • z -1
  • Next take the two equations that only have x and
    y in them and put them together. Multiply the
    first times -1 to change the signs.
  • -1(5x y 17)
  • -5x - y -17
  • 5x 4y 23
  • 3y 6
  • y 2
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