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

  • Thread starter Voivode
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In summary, the person is in a calculus course and is wondering if in the equation Ax + By = Cx + Dy, A can be equal to C and B can be equal to D. It depends on what "=" means, as if it means "identically equal to", then yes, but if it just means "equals", then no. An example is given to illustrate this distinction.
  • #1
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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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  • #2
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 [itex]\equiv[/itex] (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.
 
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  • #3
That makes sense. Thanks.
 

1. What is the basic concept behind solving equations like Ax + By = Cx + Dy?

The basic concept behind solving equations like Ax + By = Cx + Dy is to isolate the variables (A, B, C, D, x, and y) on one side of the equation and the constants on the other side. This allows us to find the value of the variables and solve the equation.

2. How do I solve for x and y in equations like Ax + By = Cx + Dy?

To solve for x and y in equations like Ax + By = Cx + Dy, you can use the properties of equality, such as combining like terms, distributing, and factoring. You can also use the substitution method or the elimination method to solve the equations.

3. What does it mean if A=C and B=D in the equation Ax + By = Cx + Dy?

If A=C and B=D in the equation Ax + By = Cx + Dy, it means that the coefficients of the x and y variables on both sides of the equation are equal. This can be helpful in solving the equation, as it allows us to simplify the equation and eliminate one variable.

4. Can I use the same method to solve equations with different coefficients?

Yes, you can use the same methods to solve equations with different coefficients. However, the steps and calculations may vary depending on the specific coefficients and variables in the equation.

5. Are there any special cases or exceptions when solving equations like Ax + By = Cx + Dy?

Yes, there are a few special cases or exceptions when solving equations like Ax + By = Cx + Dy. For example, if A=C and B=D, the equation can have infinitely many solutions. If A=C but B≠D, the equation has no solution. And if A=C=0 and B=D=0, the equation has infinitely many solutions as well. It's important to consider these special cases when solving equations.

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