A First Course in Linear Algebra, 2017a

1.2. Systems of Equations, Algebraic Procedures 23
1
3 times the second row to the first row. ⎡
⎣
1 0 −5 3
0 1 −10 0
0 0 0 0
This is in reduced row-echelon form, which you should verify using Definition 1.13. The equations
corresponding to this reduced row-echelon form are
⎤
⎦
or
x − 5z = 3
y − 10z = 0
x = 3 + 5z
y = 10z
Observe that z is not restrained by any equation. In fact, z can equal any number. For example, we can
let z = t, where we can choose t to be any number. In this context t is called a parameter . Therefore, the
solution set of this system is
x = 3 + 5t
y = 10t
z = t
where t is arbitrary. The system has an infinite set of solutions which are given by these equations. For
any value of t we select, x,y, andz will be given by the above equations. For example, if we choose t = 4
then the corresponding solution would be
x = 3 + 5(4)=23
y = 10(4)=40
z = 4
In Example 1.22 the solution involved one parameter. It may happen that the solution to a system
involves more than one parameter, as shown in the following example.
Example 1.23: A Two Parameter Set of Solutions
Find the solution to the system
x + 2y − z + w = 3
x + y − z + w = 1
x + 3y − z + w = 5
♠
Solution. The augmented matrix is ⎡
⎤
1 2 −1 1 3
⎣ 1 1 −1 1 1 ⎦
1 3 −1 1 5
We wish to carry this matrix to row-echelon form. Here, we will outline the row operations used. However,
make sure that you understand the steps in terms of Algorithm 1.19.