Homework 9
For the following four problems, use he method of elimination to determine
whether the given linear system is consistent or inconsistent. For each consistent
system, nd the solution if it is unique; otherwise, describe the innite solution
in terms of an arbitrary parameter t (as in Examples 5 and 7 in Chapter 3.1.
of the text).
1.
x + 3y = 9
2x + y = 8
2.
3.
4x
6x
2y = 4
3y = 7
4x + 9y + 12z =
3x + y + 16z =
2x + 7y + 3z =
4.
x
2x
x
2y +
y
y
1
46
19
z = 2
4z = 13
z = 5
For the following two problems, a second order dierential equation and its
general solution y(x) are given. Determine the constants A and B so as to nd
a solution of the dierential equation that satises the ven initial conditions
involving y(0) and y0 (0).
5.
6.
7.
y00 + 4y = 0; y(x) = A cos 2x + B sin 2x; y(0) = 3, y0 (0) = 8
y00 + 2y0 15y = 0; y(x) = Ae3x + Be 5x ; y(0) = 40, y0 (0) = 16
A system of the form
a1 x + b1 y = 0
a2 x + b2 y = 0
in which the constants on the right-hand side are all zero, is said to be
homogeneous. Explain by geometric reasoning why such a system has
either a unique solution or innitely many solutions. In the former case,
what is the unique solution?
1
In the following four problems, use elementary operations to transform each
augmented coecient matrix to echelon form if necessary. THen solve the system by back substitution.
8.
9.
10.
11.
x1
x1 + x2
x2
2x3 + x4 = 9
x3 + 2x4 = 1
x3
3x4 = 5
5x2 +
2x3
7x4 + 11x5 = 0
13x3 + 3x4
7x5 = 0
x4
5x5 = 0
x2
3x1 + x2
3x3 =
2x1 + 7x2 + x3 =
2x1 + 5x2
=
6
9
5
2x1 + 5x2 + 12x3 = 6
3x1 + x2 + 5x3 = 12
5x1 + 8x2 + 21x3 = 17
12. Determine for what values of k the system
3x + 2y = 1
6x + 4y = k
has (a) a unique solution, (b) no solution, and (c) innitely many solutions.
2