Linear Algebra, 2020a

10 Chapter One. Linear Systems
̌ 1.19 Use Gauss’s Method to solve each system or conclude ‘many solutions’ or ‘no
solutions’.
(a) 2x + 2y = 5
x − 4y = 0
(b) −x + y = 1
x + y = 2
(c) x − 3y + z = 1
x + y + 2z = 14
(d) −x − y = 1
−3x − 3y = 2
(e) 4y + z = 20
2x − 2y + z = 0
x + z = 5
x + y − z = 10
(f) 2x + z + w = 5
y − w =−1
3x − z − w = 0
4x + y + 2z + w = 9
1.20 Solve each system or conclude ‘many solutions’ or ‘no solutions’. Use Gauss’s
Method.
(a) x + y + z = 5 (b) 3x + z = 7 (c) x + 3y + z = 0
x − y = 0 x − y + 3z = 4 −x − y = 2
y + 2z = 7 x + 2y − 5z =−1 −x + y + 2z = 8
̌ 1.21 We can solve linear systems by methods other than Gauss’s. One often taught
in high school is to solve one of the equations for a variable, then substitute the
resulting expression into other equations. Then we repeat that step until there
is an equation with only one variable. From that we get the first number in the
solution and then we get the rest with back-substitution. This method takes longer
than Gauss’s Method, since it involves more arithmetic operations, and is also
more likely to lead to errors. To illustrate how it can lead to wrong conclusions,
we will use the system
x + 3y = 1
2x + y =−3
2x + 2y = 0
from Example 1.13.
(a) Solve the first equation for x and substitute that expression into the second
equation. Find the resulting y.
(b) Again solve the first equation for x, but this time substitute that expression
into the third equation. Find this y.
What extra step must a user of this method take to avoid erroneously concluding a
system has a solution?
̌ 1.22 For which values of k are there no solutions, many solutions, or a unique
solution to this system?
x − y = 1
3x − 3y = k
1.23 This system is not linear in that it says sin α instead of α
2 sin α − cos β + 3 tan γ = 3
4 sin α + 2 cos β − 2 tan γ = 10
6 sin α − 3 cos β + tan γ = 9
and yet we can apply Gauss’s Method. Do so. Does the system have a solution?
̌ 1.24 What conditions must the constants, the b’s, satisfy so that each of these
systems has a solution? Hint. Apply Gauss’s Method and see what happens to the
right side.