# Why are we allowed to add two or more linear equations when solving for a variable?

1. Dec 31, 2009

### 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?

2. Dec 31, 2009

### Fredrik

Staff Emeritus
Re: Why are we allowed to add two or more linear equations when solving for a variabl

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.

3. Dec 31, 2009

### Juwane

Re: Why are we allowed to add two or more linear equations when solving for a variabl

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

4. Dec 31, 2009

### Pengwuino

Re: Why are we allowed to add two or more linear equations when solving for a variabl

Yes, by the same argument.

5. Dec 31, 2009

### Juwane

Re: Why are we allowed to add two or more linear equations when solving for a variabl

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?

6. Dec 31, 2009

### pbandjay

Re: Why are we allowed to add two or more linear equations when solving for a variabl

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

7. Jan 1, 2010

### elibj123

Re: Why are we allowed to add two or more linear equations when solving for a variabl

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

8. Jan 1, 2010

### Fredrik

Staff Emeritus
Re: Why are we allowed to add two or more linear equations when solving for a variabl

Yes, if c and d are ≠0.