My question is:
In the above image, does the minus sign before the fraction represent that all the values, i.e, 2x and y, will be -2x and -y? Or does it simply imply that the values will be -2x and +y? i.e, y will remain unchanged.
Similarly, can you please tell me what the value of the above calculation will be? Also please explain how you deduced the answer.
Thank you. This question might seem simple but it is really confusing me.
On
Yes, the negative will be distributed to the numerator only and then it will change to a plus.
And so you will get, x + (-2x -y) / 3 = 1
But the way it is written right now I would leave it unless you need to solve for one of your variables (x or y).
As for the second equation, you have a number (-1) minus a fraction. You first need to get a common denominator before you can combine and subtract the two terms. So, multiply -1 by 2/2. This is allowed in math and used often because you are multiplying by 1 essentially. In doing so you don't change the equation you just manipulate it to look different while keeping the equality. And so you get,. x = -2/2 - 1/2. This further simplifies by combining the numerators and writing it all over the one common denominator: x = -3/2
No, for example the fraction $-\frac12$ is the same as $\frac{-1}{2}$ and also the same as $\frac{1}{-2}$.
The first, $-\frac12$, says divide $1$ by $2$ and make the result negative.
The second, $\frac{-1}{2}$, says divide $(-1)$ by $2$.
The third, $\frac{1}{-2}$, says divide $1$ by $(-2)$
The result of all three is the same: $-0.5$.
It would be a mistake to say $-\frac12=\frac{-1}{-2}$ because the left side is negative and the right side is positive.
So the negative is applied to the numerator, or the denominator, but not to both.
When you apply the negative to the numerator, you are multiplying the whole numerator by $(-1)$ and you must use the distributive property.
$(-1)\cdot(2x+y)=-2x-y$
The value of $-1-\frac12$ is equivalent to $-\frac22-\frac12=-\frac32$.