Page 324 - 35Linear Algebra
P. 324
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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