System of Linear Equations - Proving

[lkh1986]

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.

 

[verty]

Hint: Think about what it would mean for [ax + by = 0, cx + dy = 0] to have no more than 1 solution.

 

[Mark44]

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.

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

Thanks.