Question

True or false and explain why?

A system of 2 equations in 3 unknown has infinitely many
solutions.

Answer #1

Is each statement true or false? If true, explain why; if false,
give a counterexample.
a) A linear system with 5 equations and 4 unknowns is always
inconsistent.
b) If the coefficient matrix of a homogeneous system has a
column of zeroes, then the system has infinitely many solutions.
(Note: a homogeneous system has augmented matrix [A | b] where b =
0.)
c) If the RREF of a homogeneous system has a row of zeroes, then
the system has...

Geometrically, why does a homogenous system of two linear
equations in three variables have infinitely many solutions? If the
system were nonhomogeneous, how many solutions might there be?
Explain this geometrically.

Find the value (s) of for which the system of equations below
has infinitely many solutions. ( a-3)x+y=0 x+( a-3 ) y=0

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)= ?

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) =

For a real number "a", consider the system of equations:
x+y+z=2
2x+3y+3z=4
2x+3y+(a^2-1)z=a+2
Which of the following statements is true?
A. If a= 3 then the system is inconsistent.
B. If a= 1 then the system has infinitely many solutions.
C. If a=−1 then the system has at least two distinct
solutions.
D. If a= 2 then the system has a unique solution.
E. If a=−2 then the system is inconsistent.

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

The augmented matrix represents a system of linear equations in
the variables x and y.
[1 0 5. ]
[0 1 0 ]
(a) How many solutions does the system have: one, none, or
infinitely many?
(b) If there is exactly one solution to the system, then give
the solution. If there is no solution, explain why. If there are an
infinite number of solutions, give two solutions to the system.

true/false : if a system of linear equations has a 3x5 augmented
matrix whose fifth column has a leading term, then the system is
consistent.
I know the answer is false, but why?
This is all the information provided in the question.

Let A be a n × n matrix, and let the system of linear equations
A~x = ~b have infinitely many solutions. Can we use Cramer’s rule
to find x1? If yes, explain how to find it. If no, explain why
not.

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