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## Main Question or Discussion Point

Suppose we have a system of 2 equations in 2 unknowns x,y:

a

a

If the determinant of

[a1 b1

a2 b2]

is nonzero, then the solution to the system exists and is unique. [I am OK with this]

If the determinant of

[a1 b1

a2 b2]

is zero,

this does not distinguish between the cases of no solution and infinitely many solutions.

To gain some insight, we need to further check the determinatnts of

[a1 c1

a2 c2]

and

[b1 c1

b2 c2]

===================================

Is the bolded part correct or not? I don't see why it is true. How can we prove it?

Thanks for any help!

a

_{1}x+b_{1}y=c_{1}a

_{2}x+b_{2}y=c_{2}If the determinant of

[a1 b1

a2 b2]

is nonzero, then the solution to the system exists and is unique. [I am OK with this]

If the determinant of

[a1 b1

a2 b2]

is zero,

this does not distinguish between the cases of no solution and infinitely many solutions.

To gain some insight, we need to further check the determinatnts of

[a1 c1

a2 c2]

and

[b1 c1

b2 c2]

**If both determinants are zero, then the system has infinitely many soltuions.**

If at least one of the two determinants are nonzero, then the system has no solution.If at least one of the two determinants are nonzero, then the system has no solution.

===================================

Is the bolded part correct or not? I don't see why it is true. How can we prove it?

Thanks for any help!

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