Page 324 - 35Linear Algebra

324 Sample First Midterm

Use your result to solve the system

x + y − z + 2w = 7




x + 3y + 2z + 2w = 6

 −x − 3y − 4z + 6w = 12


4y + 7z − 2w = −7

Solutions
1. As an additional exercise, write out the row operations above the ∼ signs
below.

     3 11 
1 3 0 4 1 3 0 4 1 0
5 5
 .
 1 −2 1 1  ∼  0 −5 1 −3  ∼  0 1 − 1 5 3 





5
2 1 1 5 0 −5 1 −3 0 0 0 0
Solution set is
11
     3  
 x 5 − 5 
y
  =  3 + µ  1  : µ ∈ R .
5 5
z 0 1
 
11
 
5
3
Geometrically this represents a line in R through the point  3 running
5
0
 3 
−
5
.
 1 
parallel to the vector  5
1
 11   3 
5 − 5
The vector  3  is a particular solution and  1  is a homogeneous
5 5
0 1
solution.
As a double check note that
11
         3   
1 3 0 4 1 3 0 − 0
5 5
1 −2 1 3 = 1 and 1 −2 1 1 = 0 .
           
5 5
2 1 1 0 5 2 1 1 1 0
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