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Int. Alg. Notes
Section 4.3
Page 1 of 7
Section 4.3: Systems of Linear Equations in Three Variables
Big Idea: A system of three linear equations in three unknowns is the next level of difficulty in solving systems
of linear equations. We will use elimination to get the system into triangular form, which is then very simple to
solve.
Big Skill: You should be able to convert a system of equations to triangular form using the method of
elimination, and then solve it using back substitution. This is called Gaussian elimination.
Example of a system of three linear equations in three unknowns:
y  z  2
 x 

 x  2 y  3z  12
2 x  2 y  z  9

A linear equation with three variables describes a two-dimensional plane embedded in three dimensions:
When two planes intersect, the intersection is usually a line and when three planes intersect, the intersection is
usually a point:
y
x
Practice:
1. Verify (-1, 2, -3) is the solution to the above system.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 2 of 7
Geometric/Visual Interpretation of a System of Three Linear Equations in Three Variables:
Exactly one solution: A consistent system with independent equations where the planes intersect at a single
point. Picture:
No solution: An inconsistent system where the planes are either all parallel, or intersect along parallel lines.
Inconsistent systems yield a false equation (like 0 = 3) after trying to solve them. Picture:
Infinitely many solutions: A consistent system with dependent equations where the planes all intersect along
the same line, or are all coincident. Dependent systems yield one or more equations of 0 = 0 after applying
Gaussian elimination. Picture:
Example of a system of three linear equations in three unknowns that is in triangular form:
x  2 y  z  1

y  2z  5


z  3

Notice: the name triangular form comes from the “blank” triangular space in the lower left corner due to no x
or y variables. Also, this system is really easy to solve using back-substitution:
y  23  5
y65
y  1
x  2y  z 1
x  2   1  3  1
x 1  1
x2
The goal of this section is to learn how to convert any given system of linear equations into triangular form. We
will do this using the method of elimination, which involves adding constant multiples of equations together to
eliminate variables.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 3 of 7
Key fact behind the technique of elimination:

Multiplying an equation in a system by a constant, or adding two equations in a system together results
in a new system of equations called a transformed system, and the solution to the transformed system is the
same as the solution to the original system.
Steps for Solving a System of Three Linear Equations in Three Unknowns Using Elimination

Eliminate the x variable from the second and third equations using elimination.

Eliminate the y variable from the third equation using elimination.

Solve for z in the third equation.

Substitute z into the second equation to find the solution for y, then substitute y and z into the first
equation to find the solution for x.
Rules for showing your work:

Draw an arrow from one transformed system to the next, and write on the arrow what you did to
transform the system.

Any equation that is unchanged gets copied from one system to the next.
Example:
y  z  2
 x 

 x  2 y  3 z  12
2 x  2 y  z  9

 (1)(eqn. #1) is placed in row#1
 2 x  2 y  2 z  4

y  4 z  14


 4y 
z  5

 (4)(eqn. #2) is placed in row#2
y  z 
2
 x 

 x  2 y  3 z  12
 2 x  2 y  z  9

 (eqn. #1)  (eqn. #2) is placed in row#2
 2 x  2 y  2 z  4

4 y  16 z  56


 4y 
z  5

 (eqn. #2)  (eqn. #3) is placed in row#3
y 
z 
2
 x 

y  4 z  14

 2x  2 y 
z  9

 2 x  2 y  2 z  4

4 y  16 z  56


 17 z  51

 (eqn. #3)  ( 17) is placed in row#3
 (2)(eqn. #1) is placed in row#1
 2 x  2 y  2 z  4

y  4 z  14

 2x  2 y 
z  9

 (eqn. #1)  (eqn. #3) is placed in row#3
 2 x  2 y  2 z  4

y  4 z  14


 4y 
z  5

 2 x  2 y  2 z  4

4 y  16 z  56


z  3

Substitute into equation #2:
4 y  16  3  56  y  2
Substitute into equation #1:
2 x  2  2   2  3  4  x  1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 4 of 7
Practice:
y  z  6
 x 

2. Solve 3x  2 y 
z  5 using elimination.
 x  3 y  2 z  14

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 5 of 7
 3z  3
 x

3. Solve 
3 y  4 z  5 using elimination.
3x  2 y
 6

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 6 of 7
 2 x  2 y  3z  6

4. Solve  4 x  3 y  2 z  0 using elimination.
2 x  3 y  7 z  1

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 4.3
Page 7 of 7
y  z 
1
 x 

5. Solve  x  2 y  3z  4 using elimination.
 3x  2 y  7 z  0

Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.