Add/Subtract Linear Equations to Solve for Variable

[Juwane]

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?

 

[Fredrik]

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.

 

[Juwane]

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

 

[Pengwuino]

Yes, by the same argument.

 

[Juwane]

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?

 

[pbandjay]

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.

 

[elibj123]

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.

 

[Fredrik]

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.