A First Course in Linear Algebra, 2017a

42 Systems of Equations
Exercises
Exercise 1.2.1 Find the point (x 1 ,y 1 ) whichliesonbothlines,x+ 3y = 1 and 4x − y = 3.
Exercise 1.2.2 Find the point of intersection of the two lines 3x + y = 3 and x + 2y = 1.
Exercise 1.2.3 Do the three lines, x + 2y = 1,2x − y = 1, and 4x + 3y = 3 have a common point of
intersection? If so, find the point and if not, tell why they don’t have such a common point of intersection.
Exercise 1.2.4 Do the three planes, x + y − 3z = 2, 2x + y + z = 1, and 3x + 2y − 2z = 0 have a common
point of intersection? If so, find one and if not, tell why there is no such point.
Exercise 1.2.5 Four times the weight of Gaston is 150 pounds more than the weight of Ichabod. Four
times the weight of Ichabod is 660 pounds less than seventeen times the weight of Gaston. Four times the
weight of Gaston plus the weight of Siegfried equals 290 pounds. Brunhilde would balance all three of the
others. Find the weights of the four people.
Exercise 1.2.6 Consider the following augmented matrix in which ∗ denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is
the solution unique?
⎡
⎢
⎣
∗ ∗ ∗ ∗ ∗
0 ∗ ∗ 0 ∗
0 0 ∗ ∗ ∗
0 0 0 0 ∗
⎤
⎥
⎦
Exercise 1.2.7 Consider the following augmented matrix in which ∗ denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is
the solution unique?
⎡
⎤
∗ ∗ ∗
⎣ 0 ∗ ∗ ⎦
0 0 ∗
Exercise 1.2.8 Consider the following augmented matrix in which ∗ denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is
the solution unique?
⎡
⎢
⎣
∗ ∗ ∗ ∗ ∗
0 0 ∗ 0 ∗
0 0 0 ∗ ∗
0 0 0 0 ∗
⎤
⎥
⎦
Exercise 1.2.9 Consider the following augmented matrix in which ∗ denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is consistent. If consistent, is