Solving Linear Equation Systems: Intersection & Geometric Interpretation

In summary, the conversation discusses determining whether two given planes are parallel or intersecting, finding the equation of the line of intersection if they do intersect, and interpreting the geometric meaning of this system of equations. The solution involves realizing that two planes cannot intersect at a single point, and therefore finding the line of intersection by setting the two equations equal to each other and solving for the variables. This results in parametric equations for the line of intersection.
  • #1
kaybaby
3
0

Homework Statement


Determine whether the following planes are parallel or intersect. If they intersect, find the equation of the line of intersection. Interpret this system of two linear equations geometrically.

4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0


Homework Equations





The Attempt at a Solution


I've shown that the plane intersect at one point, and calculated the direction vector, as well as the the two normal vectors. how am i supposed to find the equation of the line of intersection if i do not get a point?
 
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  • #2
have a think geometrically - how can 2 planes intersect in a single point?
 
  • #3
kaybaby said:

Homework Statement


Determine whether the following planes are parallel or intersect. If they intersect, find the equation of the line of intersection. Interpret this system of two linear equations geometrically.

4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0
These are the same as z= 4x- 3y- 1 and z= -2x- 4y+ 5. On their line of intersection (as lanedance implies, two planes cannot intersect at a single point), z= 4x- 3y- 1= 2x- 4y+ 5. You can solve that for y as a linear function of x, then put that back into either equation to get z as a linear function of x. Set x= t and you have parametric equations for the line of intersection.


Homework Equations





The Attempt at a Solution


I've shown that the plane intersect at one point, and calculated the direction vector, as well as the the two normal vectors. how am i supposed to find the equation of the line of intersection if i do not get a point?
 

1. What is the purpose of solving linear equation systems?

The purpose of solving linear equation systems is to find the values of the variables that satisfy all of the equations in the system. This can help us understand the relationships between multiple variables and make predictions or solve problems in various fields such as physics, engineering, and economics.

2. What is the difference between intersection and geometric interpretation of linear equation systems?

The intersection of two or more linear equations is the point where they intersect on a graph. This point represents the solution to the system. Geometric interpretation, on the other hand, involves visualizing the equations as lines in a coordinate plane and understanding how they relate to each other and the coordinates of the solution point.

3. How do you solve a linear equation system algebraically?

To solve a linear equation system algebraically, we use methods such as substitution, elimination, and graphing. These involve manipulating the equations to eliminate one variable and solve for the other, or finding the intersection point of the lines on a graph.

4. Can you solve a linear equation system with more than two variables?

Yes, it is possible to solve a linear equation system with more than two variables. However, the number of equations must be equal to or greater than the number of variables in order to have a unique solution. Otherwise, the system is considered to be underdetermined or overdetermined and may have infinitely many solutions or no solution at all.

5. What are some real-life applications of solving linear equation systems?

Solving linear equation systems has many real-life applications, such as in budget planning, optimizing production processes, and predicting market trends. It is also used in fields such as engineering to design structures and systems, and in physics to model and analyze motion and other phenomena.

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