Solving Systems of Equations in Three Variables

1
Solving Systems of Equations in Three Variables

• (www.dgelman.com/)

2
Objective

• Solve a system of equations in three variables.

3
Application

• Courtney has a total of 256 points on three
  Algebra tests. His score on the first test
  exceeds his score on the second by 6 points. His
  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.
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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.

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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.

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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.

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Graphs

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

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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

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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

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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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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 ?

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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