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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