Question

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.

Answer #1

We are all too familiar with linear equations in two variables.
These systems may have no solution, one solution, or infinitely
many. Of course, we can interpret these solutions geometrically as
two parallel lines, two intersecting lines, or two identical lines
in the plane. How does this extend into linear equations in three
variables? If a linear equation in two variables describes a line,
what does a linear equation in three variables describe? Give a
geometric interpretation for the possible...

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.

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) =

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)= ?

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.

True or false and explain why?
A system of 2 equations in 3 unknown has infinitely many
solutions.

Suppose you have an n×n homogenous linear system of equations.
Make a statement classifying exactly when you have a unique
solution (what is it?) if the system is written in RREF. Prove
(i.e. explain why) your statement is true.

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

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

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.

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