Confusion about dividing the first term across the equals sign

snakehead

New member
Joined
Jul 19, 2021
Messages
4
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
Joined
Jan 16, 2018
Messages
2,252
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?
 

snakehead

New member
Joined
Jul 19, 2021
Messages
4
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
Joined
Mar 3, 2021
Messages
831
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
 

snakehead

New member
Joined
Jul 19, 2021
Messages
4
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
Joined
Mar 3, 2021
Messages
831
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
Joined
Jul 13, 2021
Messages
4
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
Joined
Apr 12, 2005
Messages
11,318
Why would you be tempted to DIVIDE BY A term? BETTER REVIEW THE DIFFEREnce BETWEEN A TERM And a FACTOR.
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
7,602
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.
 
Top