Systems of Linear Equations in Several Variables, back Substitution and
Gaussian Elimination:
Linear Equations
6 x1  3x2  5x3  10
1
x  y  z  2w 
2
Nonlinear Equations
x2  3y  z  5
x1 x1  6 x3  6
Solving a Linear System
The following are examples of systems of linear equation in three variables. The
second system is in triangular form; that is, the variable x does not appear in
the second equation and the variables x and y do not appear in the third
equation.
A system of linear equations
x  2 y  z  1

 x  3 y  3x  4
2 x  3 y  z  10

A system of linear
equations in triangular form
x  2 y  z  1

 y  2 z  1 or even

z3

x  2 y  z  1

 y  2z  1

2z  6

The equations in a system in triangular form does NOT have to have
its leading term’s coefficient equal to one
It is easy to solve a system that is in triangular form using back-substitution, so
our goal in these types of equations is to change our system of linear equations
to a system in triangular form.
Solving a Triangular System Using Back-Substitution
1)
Solve the system using back-substitution
x  2 y  z  1

 y  2z  5

z 3

Operations That Yield and Equivalent System:
1)
Multiply an equation by a nonzero constant
2)
Add a nonzero multiple of one equation to another
3)
Interchange the positions of two equations
To solve a linear system, we use these operations to change the system to an
equivalent triangular system. Then we use back-substitution in problem 1. This
process is called Gaussian elimination.
Steps to Solve a System of Linear Equations (Gaussian Elimination)
1)
Make one equation have 1 as the coefficient of x
2)
Use this equation to make system have only one equation with an x
3)
Eliminate a y from one of the two equations that does not have a x
in it
4)
Write the system in triangular form
5)
Use back substitution to solve the system
Again, the equations in system that is in triangular form DO NOT have to have
their leading term’s coefficient equal to one.
NOTE: With systems of 2 linear equations there were many different ways to get
an answer, with a system of 3 linear equations there are even more ways to get
an answer. The route I use in class is NOT the only route that can be used to
solve the system. YOU NEED TO SHOW WORK TO RECEIVE PARTIAL
CREDIT, WORK MUST BE NEAT!
Solving a System of Three Equations in Three Variables
2)
Solve the system using Gaussian elimination:
x  2 y  z  1

 x  y  z  4
 x  2 y  4 z  8

3)
Solve the system using Gaussian elimination:
 x  2 y  3z  1

 x  2 y  z  13
3x  2 y  5 z  3

4)
Solve the system using Gaussian elimination:
2 x  4 y  6 z  2

 x  2 y  z  13
2 x  4 y  7 z  11
