Homogeneous linear equations geometrically

In summary, homogeneous linear equations are represented as straight lines passing through the origin in a coordinate plane, making it easier to solve them. The term "homogeneous" refers to the fact that all terms have the same degree. These equations can be solved graphically by plotting points and drawing a line through the origin. They can have infinitely many solutions, and can also be represented using matrices for efficient solving.
  • #1
phymatter
131
0
what does a homogeneous linear equation in 3 variables represent geometrically ?
 
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  • #2
A plane
 
  • #3
A homogeneous linear equation is of the form ax+ by+ cz= 0 for numbers a, b, c, d. If all of a, b, c, d are 0, that is true for all (x, y, z) so the solution set is all space.

If any of a, b, c, is not 0, we can solve for that variable:
x= (-by- cz)/a
y= (-ax- cz)/b
z= (-ax- by)/c
which are planes,
 
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Related to Homogeneous linear equations geometrically

What are homogeneous linear equations geometrically?

Homogeneous linear equations are equations that can be represented geometrically as a straight line passing through the origin in a coordinate plane. This means that the solutions to these equations will always intersect at the origin.

What is the significance of the term "homogeneous" in these equations?

The term "homogeneous" refers to the fact that all the terms in the equation have the same degree, meaning they are all raised to the first power. This allows for the equation to be simplified and solved more easily.

How do you solve homogeneous linear equations geometrically?

To solve these equations geometrically, you can plot the points representing the solutions on a coordinate plane and draw a line passing through the origin. The slope of this line will be equal to the coefficient of the variable in the equation, and the y-intercept will be 0.

Can homogeneous linear equations have more than one solution?

Yes, homogeneous linear equations can have infinitely many solutions. This is because any point on the line passing through the origin can be a solution to the equation.

What is the connection between homogeneous linear equations and matrices?

Homogeneous linear equations can be represented using matrices, with the coefficients of the variables in the equation forming the entries of the matrix. This allows for efficient manipulation and solving of these equations using matrix operations.

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