# Compare these equations, why do I multiply both sides of one equation by -1

• Jul 11th 2012, 03:59 AM
au79
Compare these equations, why do I multiply both sides of one equation by -1
But not the other.

Its doing my head in. I got the problems from Khan Academy, but they weren't explained in the video or on the practice page. I don't understand the difference?

Attachment 24264Attachment 24265
the second question I would like to ask, is they have multiple choice selections, but in the answers they sometimes switch the variables around, and it has made me select the none of the above option. I always thought the answer was 2a + 2b - 4 is the answer not 2b +2a -4 , but quite often they have them the other way around, I guess thats one of the reasons I hate multiple choice. I'd rather just collect the answer for myself.
• Jul 11th 2012, 04:35 AM
emakarov
Re: Compare these equations, why do I multiply both sides of one equation by -1
Welcome to the forum. First, there is no need to write these formulas as images; they can easily be written in text form. For example, the third line of the left image says b = (1c - 5d - 3) / (-5c). It is also a good idea to learn basics of the LaTeX language for writing math formulas that is used on this site. Thus, $$b = \frac{1c - 5d - 3}{-5c}$$ gives $b = \frac{1c - 5d - 3}{-5c}$.

The left-hand side in the second line in both images is incorrect.

Concerning multiplying by -1, what one should do (multiply or not) is not a mathematical question. The proper mathematical question is "Is it true or false?" Multiplying both the nominator and the denominator by -1 does not change the value of the fraction, so the claim that $\frac{x}{y}=\frac{-x}{-y}$ is true. That's all mathematics care about. In which form to represent an answer is a matter of aesthetics, not mathematics. (This is a slight exaggeration, but it applies in this case.)

My guess is that, as a matter of style, it is preferable when neither numerator nor denominator start with a minus. In the left image, it is possible to achieve this by rewriting $\frac{c-5d-3}{-5c}$ as $\frac{3+5d-c}{5c}$. In the right image, whether we multiply by -1 or not, either the numerator or the denominator starts with a minus, so we may as well leave them as they are.

Quote:

Originally Posted by au79
the second question I would like to ask, is they have multiple choice selections, but in the answers they sometimes switch the variables around, and it has made me select the none of the above option. I always thought the answer was 2a + 2b - 4 is the answer not 2b +2a -4 , but quite often they have them the other way around.

The numbers 2a + 2b - 4 and 2b + 2a - 4 are equal. Apparently, you need to select the answer option "up to equality," not "up to the order of terms."
• Jul 11th 2012, 06:41 AM
au79
Re: Compare these equations, why do I multiply both sides of one equation by -1
Thanks for the reply emakarov, I guess I understand now. It just sucks, when you're presented with a multiple choice selection, or even worse when you type your answer in but its not to the liking of the website. So basically I was right all along.

I'm not sure what you mean the left hand side on second line is incorrect? could it be that I just short-cut the answer? basically it should have been b . -2c factoring, then should have been b = then the fraction. that was just how I did my scratch working.

Yeah with the second question, I was aware that they were equal, its just that when you're scanning a set of multiple choice answers, looking for the answer you know is right, and you don't see it, you click none of the above. I'm not sure if its a deliberate trap or not, but technically the answer should be expressed in order. Its just making it hard when you're getting questions wrong, and you can't progress through the program.
• Jul 11th 2012, 07:16 AM
emakarov
Re: Compare these equations, why do I multiply both sides of one equation by -1
Quote:

Originally Posted by au79
I'm not sure what you mean the left hand side on second line is incorrect?

The left-and side on the second line in the left image should be -5bc and in the right image -4rs - rt - 9r.

Quote:

Originally Posted by au79
Yeah with the second question, I was aware that they were equal, its just that when you're scanning a set of multiple choice answers, looking for the answer you know is right, and you don't see it, you click none of the above. I'm not sure if its a deliberate trap or not, but technically the answer should be expressed in order.

To have a unique "correct" form of an answer, one has to explicitly define such canonical form. For example, we could stipulate that in a sum, all terms with a's come before terms with b's. But then how do you order $a^2b$ and $ab^2$? It is possible to do so, but it requires laying out specific rules. Unless the website has done so, it is perfectly legitimate to ask the user to select a variant that is equal to the user's answer (provided no two provided variants are equal, of course). On the other hand, when the user has to enter an answer and the software rejects it because it is in the wrong form (e.g., because of the order of terms in a sum), this is indeed bad.