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

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In the equation Ax + By = Cx + Dy, whether A = C and B = D depends on the interpretation of the equality sign. If "=" is understood as "identically equal to," then A must equal C and B must equal D for all x and y. However, if "=" is taken as a standard equality, A and C can differ while still satisfying the equation for specific values of x and y. An example illustrates that with specific values, the equation can hold true without A equaling C or B equaling D. Thus, the equality's meaning is crucial to determining the relationship between A, B, C, and D.
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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.
 
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Voivode said:

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.
 
Last edited:
That makes sense. Thanks.
 

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