Solving Ax + By = Cx + Dy: A=C & B=D?

[Voivode]

Homework Statement

I'm in a calculus course, but this isn't really a calculus question. I was wondering if:

Ax + By = Cx + Dy

Could I say A = C and B = D?

Homework Equations

None.

The Attempt at a Solution

None.

 

[Mark44]

Homework Statement

I'm in a calculus course, but this isn't really a calculus question. I was wondering if:

Ax + By = Cx + Dy

Could I say A = C and B = D?

It depends on what "=" means. In the first equation, if "=" means $\equiv$ (identically equal to), then yes, A = C and B = D. Here, "identically equal to" means for any choice of x and y.

On the other hand, if "=" means just plain old "equals", then no.

Here's an example. Suppose x = 3 and y = 2. Your equation is equivalent to (A - C)x = (D - B)y, so if A = 5 and C = 3, and B = 1 and D = 4, then we have (5 - 3)3 = (4 - 1)2, or 6 = 6, a true statement, but A and C aren't equal, nor are B and D.

 

[Voivode]

That makes sense. Thanks.