3.4 solving systems of equations in three variables
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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3.4 Solving Systems of Equations in Three Variables
Algebra II
Mrs. Aguirre
Fall 2013
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Objective
• Solve a system of equations in three variables.
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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 Courtney’s test scores on the three tests?
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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 = Courtney’s score on the first test– Let s = Courtney’s score on the second test– Let t = Courtney’s score on the third test.
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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..
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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.
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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.
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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.
Courtney’s 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.
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Solutions?
• You know that a system of two linear equations doesn’t necessarily have a solution that is a unique ordered pair. Similarly, a system of three linear equations in three variables doesn’t 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 ✔
123
13
92
z
zy
zyx
• Solve the third equation, 3z = 123z = 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.
1423
1123
32
zyx
zyx
zyx
• 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 = 175x = 15 x = 3
Now you have x and y. Substitute values back into one of the equations that you started with.
2x – y + z = 32(3) - 2 + z = 36 – 2 + z = 34 + z = 3z = -1
1423
1123
32
zyx
zyx
zyx
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 = -175x + 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 = 113 + 3(2) -2(-1) = 11 ✔3rd 3x – 2y + 4z3(3) – 2(2) + 4(-1) = 1 ✔
1423
1123
32
zyx
zyx
zyx
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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 = 175x = 15 x = 3
Now you have x and y. Substitute values back into one of the equations that you started with.
2x – y + z = 32(3) - 2 + z = 36 – 2 + z = 34 + z = 3z = -1
1423
1123
32
zyx
zyx
zyx
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 = -175x + 4y = 23 3y = 6
y = 2