3.4
Solve Systems of Linear Equations in Three Variables
Linear equation in three variables
An
equation of the form ax+ by+ cz.= d, where a, b, and c are not all zero
System of three linear equations
A
system made up of three linear equations in three variables
Solution of a system of three linear equations
The
solution is the values of the three variables that make each equation true.
Ordered triple
A
coordinate in three variables (x, y, z)
THE ELIMINATION METHOD FOR A THREE-VARIABLE SYSTEM
Step 1 Rewrite the linear
system in three variables as a linear system in _two_ variables by using
the elimination method.
Step 2 Solve the new
linear system for both of its variables.
Step 3 Substitute the values
found in _Step 2_ into one of the original equations and solve for the
remaining variable.
If you obtain a _false_
equation, such as 0 = 1, in any of the steps, then the system has no
_solution._
If you do
not obtain a false equation, but obtain an _identity_
such as 0 = 0, then the system has _infinitely many solutions_.
Example
1
Use the elimination method
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Solve the system. |
3x - 2y
+ 4z = 20 |
Equation 1 |
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-x + 5y + 12z = 73 |
Equation 2 |
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x + 3y
- 2z
= 1 |
Equation 3 |
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1.
Rewrite the
system as a linear system in two variables. |
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3x - 2y
+ 4z = 20 |
Add _-3_ times
the third equation to the first. |
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_-11y+
10z = 17_ |
New Equation 1 |
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-x + 5y
+ 12z = 73 |
Add the third equation to the second. |
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x + 3y
- 2z
= 1_______ |
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_8y+
10z = 74_ |
New Equation 2 |
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2. Solve the new linear system for both of its
variables. |
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_-11y
+ 10z = 17_ |
Add _-1_ times
new Equation |
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2 to new Equation 1. |
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_-19y
= -57_ |
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y = _3_ |
Solve for y. |
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z = _5_ |
Substitute y in new Equation 1 or 2 and solve for z. |
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3.
Substitute y
and z into an original equation and solve. |
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x + 3y
- 2z
= 1 |
Equation 3 |
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x +_3(3) - 2(5)_= 1 |
Substitute for y and z. |
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x = _2_ |
Solve for x. |
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The
solution is the ordered triple (_2_, _3_, _5_). Check
the solution in each of the original equations. |
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Example
2
Solve a three-variable system with no solution
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Solve the system. |
2x + 4y + l0z
= 14 |
Equation 1 |
|
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|
x + 2y
+ 5z = -4 |
Equation 2 |
|
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3x - 4y
- 3z
= 15 |
Equation 3 |
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2x + 4y + l0z
= 14 |
Add _-2 _ times the
second |
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equation to the
first. |
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_0 = 22_ |
New Equation 1 |
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Because you obtain a _false
equation_, you can conclude that the original system has _no solution_. |
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Example
3
Solve a three-variable system with many solutions
|
Solve the system. |
2x - 2y
+ 4z = 6 |
Equation 1 |
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4x + 2y + 8z = 12 |
Equation 2 |
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4x - 2y
+ 8z = 12 |
Equation 3 |
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1.
Rewrite the system
as a linear system in two variables. |
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2x - 2y
+ 4z = 6 |
Add the first equation |
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_4x
+ 2y + 8z = 12_ |
to the
second. |
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_6x
+ 12z = 18_ |
New Equation 1 |
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4x + 2y + 8z
= 12 |
Add the second equation |
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4x - 2y
+ 8z = 12 |
to the
third. |
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_8x + 16z = 24_ |
New Equation 2 |
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2.
Solve the new
linear system for both of its variables. |
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24x + 48z =
72 |
Add _4_ times new Equation 1 |
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and -3__times new |
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_0 = 0_ |
Equation 2. |
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Because
you obtain the identity _0_ = _0_, the system has _infinitely
many solutions_. |
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3.
Describe the
solution. One way to do this is to divide new Equation 1 by 6 to get _x
+ 2z = 3_ or x = _-2z+ 3_. Substituting
this into original Equation 1 producesy = _0_ So, any ordered triple of the form (_-2z+ 3_, _0_, _ z_) is
a solution of the system. |
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Example
4 Solve a system using substitution |
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Solve the
system. |
2x + y + z = 8 |
Equation 1 |
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-x + 3y
- 2z
= 3 |
Equation 2 |
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y = x +
z |
Equation 3 |
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1. Rewrite the system as a linear system in
two variables by substituting x + z for y in Equations 1
and 2. |
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2x + y + z = 8 |
Write
Equation 1. |
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2x + ( x + z ) + z = 8 |
Substitute
for y. |
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_3x + 2z_= 8_ |
New
Equation 1 |
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-x + 3y - 2z = 3 |
Write
Equation 2. |
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-x + 3( x + z ) - 2z = 3 |
Substitute
for y. |
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_2x+ z_ = _3_ |
New
Equation 2 |
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2. Solve the new linear system in two variables. |
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_3x + 2z = 8_ |
Add new
Equation 1 and |
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_-4x - 2z
= -6_ |
-2 times
new Equation 2. |
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x = _-2_ |
Solve for x. |
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z = _7_ |
Substitute
into new Equation 1 or new Equation 2 to find z |
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y = _5_ |
Substitute
into an original equation to find y. |
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The solution is ( _-2_, _5
_, _7_ ). |
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