Download Determine Whether an Ordered Triple is a Solution of a System

Math 35
3.4 "Systems of Three Linear Equations in Three Variables"
Objectives:
* Determine whether an ordered triple is a solution of a system.
* Solve systems of equations in three variables.
Preliminaries:
In previous sections, we solved systems of linear equations in two variables. We will now extend this discussion to
consider systems of linear equations in three variables.
Determine Whether an Ordered Triple is a Solution of a System
De…nition:
"Standard (General) Form"
A linear equation in three variables is an equation that can be written in the form
:
where A; B; C; and D are real numbers and A; B; and C are not all zero.
Example 1: (Checking for solutions)
Determine whether (6; 3; 1) is a solution of the system:
8
>
< x y + z = 10
x + 4y z = 7
>
:
3x y + 4z = 24
Solve Systems of Three Linear Equations in Three Variables
Solving a system of Three Linear Equations by Elimination:
1. Write the equations in standard form (where A; B; C; and D are integers).
2. Pick any two equations and eliminate a variable.
3. Pick a di¤erent pair of equations and eliminate the same variable as in step (2) .
4. Solve the resulting pair of two equations in two variables.
5. To …nd the third variable, substitute the values found in step 4 into any original equation.
6. Write the solution as an ordered triple (and check the solution).
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Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
3.4
Example 2: (Consistent system)
Solve
1equations:
0 the following system of
2x y + 3z = 14
C
B
C
B
a) B 4x + 2y z = 12
C
A
@
6x 3y + 4z = 22
0
3x + 2y
B
B
b) B 2x
@
4x
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2z = 14
1
C
C
5y + 3z = 7 C
A
3y + 7z = 5
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
3.4
Solve Systems of Equations with Missing Variable Terms
Example 3: (Solving systems with missing variable(s))
Solve
8 the following systems.
>
>
x + 2y = 1 + z
>
<
a)
2x = 3 + y z
>
>
>
: x+z =3
8
>
>
x + y 4z =
>
<
b)
x y= 6
>
>
>
: 3y + z = 12
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54
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
3.4
Identify Inconsistent Systems
Example 4: (Inconsistent systems)
8
>
> 2a + b 3c = 8
>
<
Solve the system (if possible):
3a 2b + 4c = 10
>
>
>
: 4a + 2b 6c = 5
Example 5: (Consistent
8 systems with in…nitely many solutions)
>
> x y=3
>
<
Solve the system:
2x y + z = 1
>
>
>
: x+z = 2
In summary we have:
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Bibiana Lopez