System of Linear Equations - Proving

In summary, the conversation discusses the relationship between the solutions of two different systems of equations with the same variables. It is shown that if the first system has two different solutions, the second system will also have two different solutions. The conversation also mentions the different cases in which two straight lines can intersect and how this relates to the number of solutions for a system of equations. Finally, it is suggested to use row equivalent or equivalent matrices to prove the relationship between the two systems of equations.
  • #1
lkh1986
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0

Homework Statement



Given that the values for a, b, c, d, e and f for the system ax+by=e, cx+dy=f has two different solutions. Show that ax+by=0, cx+dy=0 also has two different solutions.

Homework Equations





The Attempt at a Solution



There're three cases of how two straight lines can intersect:
(i) At only 1 point: unique solution
(ii) They are parallel and do not intersect: no solution
(iii) They are the same line: infinitely many solutions

I assume by two, it means infinitely many solutions?

Hence, the matrix [a b; c d] is singular? I think I need to use row equivalent or equivalent matrix for this?

Thanks.
 
Last edited:
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  • #2
Hint: Think about what it would mean for [ax + by = 0, cx + dy = 0] to have no more than 1 solution.
 
  • #3
lkh1986 said:

Homework Statement



Given that the values for a, b, c, d, e and f for the system ax+by=e, cx+dy=f has two different solutions. Show that ax+by=0, cx+dy=0 also has two different solutions.

Homework Equations





The Attempt at a Solution



There're three cases of how two straight lines can intersect:
(i) At only 1 point: unique solution
(ii) They are parallel and do not intersect: no solution
(iii) They are the same line: infinitely many solutions

I assume by two, it means infinitely many solutions?
Yes, because two straight lines can't intersect in exactly two points.
lkh1986 said:
Hence, the matrix [a b; c d] is singular? I think I need to use row equivalent or equivalent matrix for this?

Thanks.
 

FAQ: System of Linear Equations - Proving

What is a system of linear equations?

A system of linear equations is a set of two or more equations that involve two or more variables. The solutions to the equations are the values of the variables that make all of the equations true at the same time.

How can I prove that a system of linear equations has a unique solution?

A system of linear equations has a unique solution if the number of equations is equal to the number of variables, and the system is consistent (meaning there is at least one solution) and independent (meaning none of the equations can be obtained by multiplying or adding the other equations).

What is the process for proving a system of linear equations using the substitution method?

The substitution method involves solving one of the equations for one of the variables, and then substituting that expression into the other equations. This will create a new system of equations with one less variable, which can then be solved using elimination or substitution again.

Can a system of linear equations have no solution?

Yes, a system of linear equations can have no solution. This occurs when the equations are inconsistent, meaning there is no set of values that will satisfy all of the equations at the same time.

How can I check my solution to a system of linear equations?

To check your solution, plug the values into each equation and see if they make the equation true. If the values make all of the equations true, then the solution is correct. You can also graph the equations and see if the solution point falls on all of the lines.

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