Question

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 solutions of a 3x3 linear system

Answer #1

Equation in three variable describes the equation of a plane in 3 dimensions.

It's equation is = aX + b.Y + c.Z = d

Where a ,b ,c,d are constants and are called direction cosines of a normal vector to the plane.

Like as line these system mau have no solution as if planes are parallel

If planes are intersecting then intersection point is a line

And if planes are coincidence then infinite no. Of solutions exist.

Geometrical interpretation of a 3 variable equation is a geometrical plane.

Plane is a surface having minimum two line containing in it.

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.

The augmented matrix represents a system of linear equations in
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[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
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Please give examples of linear systems which
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(2) has 4 variables and 2 equations and is inconsistent.
(3) has 4 variables and 2 equations and has exactly one
solution.
(4) has 4 variables and 2 equations and has infinite many
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Two systems of equations are given below. For each system,
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-x + 2y = -4
x- 2y = 4
*The system has no solution
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In Chapter 3, we have studied techniques for solving linear
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In this lab we consider a two-parameter family of linear systems.
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A system of linear equations is said to be homogeneous if the
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There are three ways of solving systems of linear equations:
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5. Consider solving the following system with the addition
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Your discussion response for this unit will consist of two
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First, create 3 equations of the form , where a, b, c,
and d are constants (integers between – 5 and 5). For
example, . Perform row operations on your system to obtain a
row-echelon form and the solution.
Go to the 3D calculator website GeoGebra:
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After you have completed this first task, choose
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1.) You will work with 0.10 M acetic acid and 17 M acetic acid
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