Question

For each part below, give an example of a linear system of
**three equations in three variables** that has the
given property. in each case, explain how you got your answer,
possibly using sketches.

(a) has no solutions

(b) has exactly one solution which is (1, 2, 3).

(c) any point of the line given parametrically be (x, y, z) = (s − 2, 1 + 2s, s) is a solution and nothing else is.

(d) any point of the form (x, y, z) = (s1, s2, s1 − s2), with s1, s2 numbers, is a solution and nothing else is. Hint: The set of all such points is a plane in 3-space.

Answer #1

The augmented matrix below represents a system of linear
equations associated with a real world prob-
lem. The augmented matrix has already been completely row
reduced.
1 0 6 12 0 1 −2 0 0000
(a) Use the reduced matrix to write down the parametric solution
for the system as a point (x, y, z).
(b) Assuming that x, y, and z represent the number of whole
items, determine how many “actual” solutions this system has, and...

4. [10] Consider the system of linear equations
x + y + z = 4
x + y + 2z = 6
x + y + (b2 − 3)z = b + 2
where b is an unspecified real number. Determine, with
justification, the values of b (if any) for which the system
has
(i) no solutions;
(ii) a unique solution;
(ii) infinitely many solutions.

Find the values of a and b for which the following system of
linear equations is (i) inconsistent; (ii) has a unique solution;
(iii) has infinitely many solutions. For the case where the system
has infinitely many solutions, write the general solution.
x + y + z = a
x + 2y ? z = 0
x + by + 3z = 2

1)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express your answer in terms
of the parameters t and/or s.)
x1
+
2x2
+
8x3
=
6
x1
+
x2
+
4x3
=
3
(x1,
x2, x3)
=
2)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express...

Determine the value of k such that the following system of
linear equations has infinitely many solutions, and then find the
solutions. (Express x, y, and z in terms of the parameters t and
s.) 3x − 2y + 4z = 9 −9x + 6y − 12z = k k = (x, y, z) =

Solve the system of linear equations. If the system has an
infinite number of solutions, set w = t and solve for x, y, and z
in terms of t.)
x + y + z + w = 6
2x+3y - w=6
-3x +4y +z + 2w= -1
x + 2y - z + w = 0
x, y, z, w=?

Solve the system of equations.
4x-3y+z = 18
x+y = 7
Select the correct choice below and, if necessary, fill in the
answer boxes to complete your choice.
A. This system has exactly one solution. The solution is left
parenthesis nothing comma nothing comma nothing right parenthesis .
(Type integers or simplified fractions.)
B. This system has infinitely many solutions of the form left
parenthesis nothing comma nothing comma z right parenthesis , where
z is any real number. (Type...

Exercise 2.4 Assume that a system Ax = b of linear equations has
at least two distinct solutions y and z.
a. Show that xk = y+k(y−z) is a solution for every
k.
b. Show that xk = xm implies k = m. [Hint:
See Example 2.1.7.]
c. Deduce that Ax = b has infinitely many solutions.

4. Solve the system of linear equations by using the
Gauss-Jordan (Matrix) Elimination Method. No credit in use any
other method. Use exactly the notation we used in class and in the
text. Indicate whether the system has a unique solution, no
solution, or infinitely many solutions. In the latter case, present
the solutions in parametric form.
3x + 6y + 3z = -6
-2x -3y -z = 1
x +2y + z = -2

A: Determine whether the system of linear equations has one and
only one solution, infinitely many solutions, or no solution.
3x - 4y = 9
9x - 12y = 18
B: Find the solution, if one exists. (If there are infinitely
many solutions, express x and y in terms of parameter t. If there
is no solution, enter no solution.)
(x,y)= ?

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