Add/Subtract Linear Equations to Solve for Variable

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Discussion Overview

The discussion revolves around the properties of linear equations, specifically focusing on the operations of addition, subtraction, multiplication, and division of equations to solve for variables. Participants explore the validity of these operations and their implications in both simple and multiple equation scenarios.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the property that allows the addition or subtraction of two different linear equations, seeking clarification on its validity.
  • Another participant explains that the equality sign indicates that both sides of an equation represent the same value, allowing for the addition of equations based on this principle.
  • A participant inquires whether the same principle applies to multiplication of equations, to which another participant affirms it does.
  • There are multiple inquiries regarding the legality of dividing equations, with one participant presenting a specific case involving rational expressions and questioning the validity of separating terms.
  • Responses indicate that division is permissible under certain conditions, specifically when the denominators are non-zero.
  • Some participants challenge the reasoning behind the division of equations, suggesting that the converse of the initial statements may not hold true.

Areas of Agreement / Disagreement

Participants generally agree on the validity of adding and multiplying equations but exhibit disagreement regarding the division of equations and the conditions under which it is valid. The discussion remains unresolved on the implications of division.

Contextual Notes

Participants express uncertainty regarding the conditions necessary for valid operations, particularly in the context of division and the implications of one-to-one matching in rational expressions.

Who May Find This Useful

This discussion may be useful for students and educators in mathematics, particularly those exploring the properties of equations and their operations in algebra.

Juwane
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When we solve for a variable in say two linear equations, by what property we are allowed to add one equation to the other or subtract one equation from the other? How can this be allowed when the two are completely different equations?

For more than two equations, does this work for adding/subtracting only two equations? Can more than two equations be simultaneously added/subtracted?
 
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The equality sign in an equation means that what you got on the two sides of it is actually the same thing, but possibly expressed in different ways. Every equation really says something like 5=5. So when you're adding two equations, you're really just saying that if a=a and b=b, then we also have a+b=a+b. This statement is of course trivially true. This holds for all equations, not just linear ones. And yes, it also holds for more than two equations, for the same reason.
 


Does this also hold for multiplication? That is, can we also multiply the two or more equations together?
 


Yes, by the same argument.
 


Can we divide them also?

If a=b and c=d, then is it true that a/c=b/d?

If the above is true, then in the case of 15/3=10/2, why can't we say 15=10 and 3=2?
 


Juwane said:
Can we divide them also?

If a=b and c=d, then is it true that a/c=b/d?

If the above is true, then in the case of 15/3=10/2, why can't we say 15=10 and 3=2?

You are trying to use the converse of your if-statement, which is not true in this case.
 


Juwane said:
Can we divide them also?

If a=b and c=d, then is it true that a/c=b/d?

If the above is true, then in the case of 15/3=10/2, why can't we say 15=10 and 3=2?

Because this separation is legal only with one-to-one matching, since the same number matches infinite number of rational presentations, the separation is illegal.
 


Juwane said:
Can we divide them also?

If a=b and c=d, then is it true that a/c=b/d?
Yes, if c and d are ≠0.
 

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