# Confusion about dividing the first term across the equals sign

##### New member
I apologise if the title is confusing (and if the rest of the post is...); this isn't necessarily a question I was given, just something I was thinking about.

why is it when my multiply the 12 to get it to the other side that the equation becomes unbalanced; why does doing the inverse not maintain the balance?:

6 = 12 / 2
6 * 12 = 1/2
[is a half even the correct product if the 12 goes to the other side?]
72 = 1/2 ??

[I know the equation is fine as it is, I just don't understand why I can't just move the 12 to the other side; what's stopping it?]

I also don't understand whether I should be multiplying; does the multiplication sign belong to the 2 or 7, or both, or neither?

Also why does it seem to work for multiplication, addition and subtraction but not division:

6 = 3 * 2
6/3 = 2

6= 1 + 5
1) 6-1 = 5
2) 6-5=1

6 = 10 - 4
1) 6-10 = -4
2) 6+4=10

I am very appreciative of your patience and time whilst you read this. I again apologise for the messiness and confusion.

#### lev888

##### Senior Member
I apologise if the title is confusing (and if the rest of the post is...); this isn't necessarily a question I was given, just something I was thinking about.

why is it when my multiply the 12 to get it to the other side that the equation becomes unbalanced; why does doing the inverse not maintain the balance?:

6 = 12 / 2
6 * 12 = 1/2
[is a half even the correct product if the 12 goes to the other side?]
72 = 1/2 ??
Let's consdier the first question. To maintain the equality you can only apply the same operation to both sides: you can either divide both sides by 12, or multiply both sides by 12. In this case you divided the right side and multiplied the left side. Makes sense?

##### New member
So to maintain equality, I can only do:
6 * 12= (12/2) * 12

or

6/12 = 1/2

But I was always taught, when you move a number from one side of the equals sign to the other, you must to the inverse operation of that number, i.e. I wanted to get rid of 12 on one side, so I multiplied it to the other. What is it that I am doing here, why does this method not apply to this equation? (apologies for the excessive questioning, very appreciative of your help).

#### lex

##### Full Member
I'm not quite sure what your method is (but putting a sign before each number)

6= 0 +1 + 5
1) 6-1 = 5
2) 6-5=1

6 = 0 +10 - 4
1) 6-10 = -4
2) 6+4=10

6 = 1 *3 * 2
6/3 = 2
6/2=3

6=1 *12 /2
6/12=1/2
6*2=12

• topsquark

##### New member
I'm not quite sure what your method is (but putting a sign before each number)

6= 0 +1 + 5
1) 6-1 = 5
2) 6-5=1

6 = 0 +10 - 4
1) 6-10 = -4
2) 6+4=10

6 = 1 *3 * 2
6/3 = 2
6/2=3

6=1 *12 /2
6/12=1/2
6*2=12
So the sign in front of 12 is multiplication. And to move it to the other side you can only do the inverse operation of a respective number's sign i.e. since 12's sign is *, I can only divide; since the 2's sign is /, I can only divide?

#### lex

##### Full Member
So the sign in front of 12 is multiplication. And to move it to the other side you can only do the inverse operation of a respective number's sign i.e. since 12's sign is *, I can only divide; since the 2's sign is /, I can only divide?
I think you meant the last word to be 'multiply'.
If so, yes, (if this is the way you think about these things), you have got it right.

#### bigjohn 2

##### New member
Snakehead.....the problem you have comes about because the secondary school system does a pathetic job of explaining the REAL NUMBER SYSTEM. Your symbol 12/ 2 means 12 times the symbol (1/2) where (1/2) is the multiplicative inverse of 2.....1/2 times 2 = 1 , the multiplicative identity . The operations of the RNS are something we call ' addition and multiplication , denoted by + & x ' . The symbol ( - 2 ) is termed the additive inverse of 2, ie (-2) + 2 = 0 .... 6 - 1 ≡ 5 + 1 + ( - 1) = 5 + 0 = 5........ 12 / 2 = 6 x 2 x (1/2) = 6 x 1 = 6

• #### tkhunny

##### Moderator
Staff member
Why would you be tempted to DIVIDE BY A term? BETTER REVIEW THE DIFFEREnce BETWEEN A TERM And a FACTOR.

#### HallsofIvy

##### Elite Member
So to maintain equality, I can only do:
6 * 12= (12/2) * 12

or

6/12 = 1/2

But I was always taught, when you move a number from one side of the equals sign to the other, you must to the inverse operation of that number, i.e. I wanted to get rid of 12 on one side, so I multiplied it to the other. What is it that I am doing here, why does this method not apply to this equation? (apologies for the excessive questioning, very appreciative of your help).
I have always disliked the phrase "move a number from one side of the equals side to the other". Students often don't understand what "moving a number" means. Rather "whatever operation you do on one side you must do on the other side". You first have "6= 12/2". To get rid of the 12 on the right side, since it is in the numerator, you need to divide by it- and, of course divide by 12 on the right: 6/12= (12/2)/12, 1/2= 1/2 Or to get rid of the 2 on the right, since it is in the denominator, you need to multiply by 2- on both sides.
6(2)= (12/2)(2), 12=12.

• tkhunny and Cubist