Determining when a system of equations has no solution and infinite solutions

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SUMMARY

The discussion focuses on determining the conditions under which the system of equations defined by ax - 2y + 3z = 5, -x + y - bz = -3, and 2x + cy - 2z = d has no solution or infinite solutions. It is established that manipulating the equations, such as multiplying the second equation by "a" and adding it to the first, can help eliminate variables and simplify the system. The participants emphasize the importance of showing progress in problem-solving to facilitate effective assistance.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly systems of equations
  • Familiarity with matrix operations and determinants
  • Knowledge of conditions for consistency and dependency in linear systems
  • Ability to manipulate algebraic expressions and equations
NEXT STEPS
  • Explore the concept of matrix rank and its relation to the number of solutions in a system of equations
  • Learn about the conditions for a system of linear equations to have no solution or infinite solutions
  • Study the method of Gaussian elimination for solving systems of equations
  • Investigate the implications of parameterized solutions in linear algebra
USEFUL FOR

Students and educators in mathematics, particularly those studying linear algebra, as well as anyone involved in solving systems of equations in engineering or applied sciences.

Ankit2
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Find the surface in terms of a,b,c on which the following system of equations has no
solution
ax-2y+3z=5
-x+y-bz=-3
2x+cy-2z=d
Could there be any values of a,b,c,d for which the system has infinite solution? (Justify).
 
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Hello and welcome to MHB! :D

I have retitled your thread so that it indicates the nature of the question being asked.

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
Ankit said:
Find the surface in terms of a,b,c on which the following system of equations has no
solution
ax-2y+3z=5
-x+y-bz=-3
2x+cy-2z=d
Could there be any values of a,b,c,d for which the system has infinite solution? (Justify).
Have you tried to solve the system? The first thing I notice is that if you multiply the second equation by "a" and add that to the first equation, you eliminate "x": (ax- 2y+ 3z)+ (-ax+ ay- abz)= (2- a)y+ (3- ab)z= 5- 3a. And that if you multiply the second equation by "2" and add that to the third equation, you also eliminate x: (2x- cy- 2z)+ (-2x+ 2y- 2bz)= (2- c)y- (2+ 2b)z= d- 6. Can you solve those two equations for y and z? If not what would stop you?
 

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