Solving Systems of Three Linear Equations in Three Variables
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Transcript of Solving Systems of Three Linear Equations in Three Variables
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Solving Systems of Three Linear Equations in Three
VariablesThe Elimination Method
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Solutions of a system with 3 equations
The solution to a system of three linear equations in three variables is an ordered triple.
(x, y, z)
The solution must be a solution of all 3 equations.
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Is (–3, 2, 4) a solution of this system?
3x + 2y + 4z = 112x – y + 3z = 45x – 3y + 5z = –1
3(–3) + 2(2) + 4(4) = 112(–3) – 2 + 3(4) = 45(–3) – 3(2) + 5(4) = –1
Yes, it is a solution to the system because it is a solution to all 3
equations.
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This lesson will focus on the Elimination Method.
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Use elimination to solve the following system of equations.
x – 3y + 6z = 213x + 2y – 5z = –302x – 5y + 2z = –6
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Step 1
Rewrite the system as two smaller systems, each containing two of the three equations.
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x – 3y + 6z = 213x + 2y – 5z = –302x – 5y + 2z = –6
x – 3y + 6z = 21 x – 3y + 6z = 213x + 2y – 5z = –30 2x – 5y + 2z = –6
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Step 2
Eliminate THE SAME variable in each of the two smaller systems.
Any variable will work, but sometimes one may be a bit easier to eliminate.
I choose x for this system.
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(x – 3y + 6z = 21) 3x + 2y – 5z = –30
–3x + 9y – 18z = –63 3x + 2y – 5z = –30
11y – 23z = –93
(x – 3y + 6z = 21) 2x – 5y + 2z = –6
–2x + 6y – 12z = –42 2x – 5y + 2z = –6
y – 10z = –48
(–3) (–2)
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Step 3
Write the resulting equations in two variables together as a system of equations.
Solve the system for the two remaining variables.
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11y – 23z = –93 y – 10z = –48
11y – 23z = –93 –11y + 110z = 528
87z = 435 z = 5
y – 10(5) = –48 y – 50 = –48
y = 2
(–11)
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Step 4
Substitute the value of the variables from the system of two equations in one of the ORIGINAL equations with three variables.
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x – 3y + 6z = 213x + 2y – 5z = –302x – 5y + 2z = –6
I choose the first equation.
x – 3(2) + 6(5) = 21x – 6 + 30 = 21 x + 24 = 21
x = –3
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Step 5
CHECK the solution in ALL 3 of the original equations.
Write the solution as an ordered triple.
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x – 3y + 6z = 213x + 2y – 5z = –302x – 5y + 2z = –6
–3 – 3(2) + 6(5) = 213(–3) + 2(2) – 5(5) = –302(–3) – 5(2) + 2(5) = –6
The solution is (–3, 2, 5).
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It is very helpful to neatly organize yourwork on your paper in the following manner.
(x, y, z)
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Solve the system.Solve the system.1.1.x+3y-z=-11x+3y-z=-112x+y+z=12x+y+z=1
z’s are easy to cancel!z’s are easy to cancel!3x+4y=-103x+4y=-10
2. 2x+y+z=12. 2x+y+z=15x-2y+3z=215x-2y+3z=21
Must cancel z’s again!Must cancel z’s again!-6x-3y-3z=-3-6x-3y-3z=-35x-2y+3z=215x-2y+3z=21 -x-5y=18-x-5y=18
2(2)+(-4)+z=12(2)+(-4)+z=1 4-4+z=14-4+z=1
3. 3x+4y=-103. 3x+4y=-10 -x-5y=18-x-5y=18
Solve for x & y.Solve for x & y.3x+4y=-103x+4y=-10-3x-15y+54-3x-15y+54
-11y=44-11y=44 y=- 4y=- 4
3x+4(-4)=-103x+4(-4)=-10 x=2x=2
(2, - 4, 1)(2, - 4, 1)
x+3y-z=-11x+3y-z=-112x+y+z=12x+y+z=1
5x-2y+3z=215x-2y+3z=21
z=1z=1
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2.2.2x+2y+z=52x+2y+z=54x+4y+2z=64x+4y+2z=6
Cancel z’s again.Cancel z’s again.-4x-4y-2z=-10-4x-4y-2z=-104x+4y+2z=64x+4y+2z=6 0=- 40=- 4
Doesn’t make Doesn’t make sense!sense!
No solutionNo solution
Solve the system.Solve the system.
1.1.-x+2y+z=3-x+2y+z=32x+2y+z=52x+2y+z=5
z’s are easy to z’s are easy to cancel!cancel!-x+2y+z=3-x+2y+z=3-2x-2y-z=-5-2x-2y-z=-5-3x=-2-3x=-2x=2/3x=2/3
-x+2y+z=3-x+2y+z=32x+2y+z=52x+2y+z=5
4x+4y+2z=64x+4y+2z=6
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3. x+y=33. x+y=32x+2y=62x+2y=6
Cancel the x’s.Cancel the x’s.-2x-2y=-6-2x-2y=-62x+2y=62x+2y=6 0=00=0
This is true.This is true.¸ ¸ many solutionsmany solutions
Solve the system.Solve the system.1.1.-2x+4y+z=1-2x+4y+z=13x-3y-z=23x-3y-z=2
z’s are easy to z’s are easy to cancel!cancel!x+y=3x+y=32.2.3x-3y-z=23x-3y-z=25x-y-z=85x-y-z=8
Cancel z’s again.Cancel z’s again.-3x+3y+z=-2-3x+3y+z=-25x-y-z=85x-y-z=82x+2y=62x+2y=6
-2x+4y+z=1-2x+4y+z=13x-3y-z=23x-3y-z=25x-y-z=85x-y-z=8
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Try this one.
x – 6y – 2z = –8–x + 5y + 3z = 23x – 2y – 4z = 18
(4, 3, –3)
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Here’s another one to try.
–5x + 3y + z = –1510x + 2y + 8z = 1815x + 5y + 7z = 9
(1, –4, 2)
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ApplicationCourtney 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 Courtney’s test scores on the three tests?
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ExploreProblems 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 problemLet f = Courtney’s score on the first testLet s = Courtney’s score on the second testLet t = Courtney’s score on the third test.
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PlanWrite 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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SolveNow 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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SolveThen 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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SolveNext 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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ExamineNow 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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GraphsThe 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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Graphs1.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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Graphs3. 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 ✔
1231392
zzyzyx
Solve the third equation, 3z = 123z = 12
z = 4Substitute 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 = 223x - 2y + 4z = 15x + 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.
1423112332
zyxzyxzyx
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
1423112332
zyxzyxzyx
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 ✔
1423112332
zyxzyxzyx
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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
1423112332
zyxzyxzyx
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